LECTURES ON DIOPHANTINE APPROXIMATIONS Part 1: gadic numbers and Roth's theorem
By
KURT MAHLER
Prepared from the notes by R.P. Bambah of my lectures given at the University of Notre Dame in the Fall of 1957
Copyright 1961 UNIVERSITY OF NOTRE DAME
FHOTOLITHOFRINTED BY GUSHING  MALLOY, ING. ANN ARBOR, MICHIGAN, UNITED STATES OF AMERICA 1961
Dedicated in Gratitude to
L. J. Mordell
My interest in two of the main subjects of these lectures goes back to my student days at Frankfurt (192325) and Gottingen (192533). Prom C. L. Siegel I learned of Thue's theorem and its improvements and generalisations; and Emmy Noether introduced me to the theory of padic numbers. I combined these two ideas in 1931 when I found an analogue of the Thue Siegel theorem that involved both real and padic algebraic numbers. In later work I repeatedly came back to such problems, and already in 193436 I gave a course on Diophantine approximations at the University of Groningen dealing with problems that simultaneously involved the real and padic fields. After the war, E. Lutz published her very beautiful little book on Diophantine approximations in the padic field. But she considered alone the case of numbers in one fixed padic field. In 1955, K. F. Roth obtained his theorem on the rational approximations of a real algebraic number. It was immediately clear that his method should also work for padic algebraic numbers, for Roth's method could clearly be combined with that of my old papers. Some interesting work of this kind was in fact carried out by D. Ridout, a student of Roth. In the second part of these lectures I shall try to go rather further in this direction. As the proofs will show, the padic numbers, and more generally the gadic numbers form an important tool in these investigations; but one form of the final result will be again free of such numbers and will state a property of rational numbers only. The first part of these lectures has mainly the purpose of acquainting the reader with the theory of padic and gadic numbers. It gives a short account of the theory of valuations, and I have found it convenient to discuss also the slightly more general theory of pseudovaluations because it leads in a very natural way to Hensel's gadic numbers. The results in Chapters 3 and 4 serve chiefly as examples, but have perhaps also a little interest in themselves. The whole second part, as well as two short appendices, deal with a very general gadic form of Roth's theorem. As the proof is rather involved, all details are given, and I have also tried to explain the reasons behind the different steps of the proof. The most original part of Roth's proof consists in a very deep theorem, here called Roth's Lemma. It states that, under certain conditions, a polynomial in a large number of variables cannot have a multiple zero of too high an order. Since Roth's Lemma is essentially a theorem on the singularities of an algebraic manifold, perhaps the methods of algebraic geometry may finally enable one to obtain a simpler proof and a stronger result (say, with the upper bound 2*n+112"^"1^ in Roth's Lemma replaced by something like tm"c). It would then become possible to improve the theorem by Cugiani given in the appendix. Another possible approach to Roth's Lemma is from the theory of
vi
LECTURES ON DIOPHANTINE APPROXIMATIONS
interpolation, or from Minkowski's theorem on the successive minima of convex bodies. As far as I know, neither of these methods has yet been applied to the problem. The method of ThueSiegelRoth has one fundamental disadvantage, that of its non effectiveness. The proof is entirely non constructive, and by its very nature does not lead to any upper bounds for possible solutions. Only in some very special cases effective methods are known and there are due to Skolem and Gelfond. Certainly the methods and results of the second part of these lectures are not the last word on the subject, and entirely new ideas are called for. So far one has not succeeded even in proving that there exist real algebraic numbers a for which the inequality
has infinitely many rational solutions Q , however small 6 > o is chosen; or that there are real algebraic numbers a. at least of the third degree for which this is not the case. I should be well pleased if these lectures succeed in convincing the reader that the whole subject is as yet in a very unsatisfactory state. Essentially in the form as printed here, I gave these lectures during the , Fall term of 1957 at the University of Notre Dame, and I wish to express my thanks to the authorities of Notre Dame for the invitation to work and lecture at this very pleasant place. My particular thanks are due to all who attended my lectures and especially to Professors Ross, Lewis, Skolem, and Bambah. In many talks with these colleagues, on the way to lunch after the lectures, much became clearer, and simpler proofs were found. I am especially indebted to R. P. Bambah who undertook the thankless task of taking down the lectures, and to T. Murphy who checked this manuscript for errors. After the work of more than a year Bauebah's notes have now at last helped me to complete this book. As the title already suggests, I hope to continue these lectures at a later date. A second part will probably deal with applications of the geometry of numbers to Diophantine approximations in padic fields. Concluded on March 12, 1959 Mathematics Department, Manchester University.
LIST OF NOTATIONS The meanings of letters and symbols will usually be clear from the context. In general, small Latin letters denote rational numbers, small Greek letters denote real or padic numbers, and capital Greek letters denote gadicor g*adic numbers. By F, P, Pp, Pg, and Pg* we mean the fields of rational, real, and padic numbers, and the rings of gadic, and g*adic numbers, respectively. The symbols ' l^olp, Wig, and U*g* stand for the absolute value of the real number a, the padic value of the padic number ao, the gadic value of the gadic number A, and the g*adic value of the g*adic number A*, respectively. Here lao'p is normed by the formula
The integer g ^ 2 always has the prime factorisation ei er g=Pl Pr > where pj , . . . , pr are distinct primes, and ej , . . . , er are positive integers. If, for j = 1, 2, . . . , r, the gadic number A has the pjadic component o?j, we write and then logg log PI
logg er log pr
max
Thus, in particular,
A g*adic number A* has, in addition to the pjadic components «j , also a real component a . We write (F (a)) is small 8. A necessary and sufficient condition for transcendency
IV.
Continued fractions 1. The continued fraction algorithm in the real case 2. The convergents of the continued fraction for  « 

•
and hence w(ama$) < e if m ^max{q(N2e),s), n ^max{q(N2e),s}. Denote by a"1 the residue class of {am}. Then a a"1 = 1, since {am} {am} = {amamK where amam = 1 if m ^ s. When w(a) is only a pseudovaluation, K^ in general will not be a field, but may contain divisors of zero. One such case will soon be discussed. 9. The limit notation.
Let again K be a field, w(a) a valuation or pseudo valuation of K, and Kw the completion of K with respect to w(a). It is convenient, and in agreement with the usual convention for the real field, to adopt the following notation. If {am} is any fundamental sequence, and if a is its residue class in KW, then we say that a is the limit of am with respect to w(a) as m tends to infinity, and we write
14
LECTURES ON DIOPHANTINE APPROXIMATIONS
a = mlim —»oo am (w). From the definition of a, this limit is naturally unique. For the fundamental sequence of the special form {a} = {a, a, a,...} we have a= lim a (w) m—>°° since this sequence lies in the residue class (a) which we have identified with the element a of K. The definition of the operations in Kw immediately implies that, if ]8 = lim bm (w) m* 0,..., Xr > 0. An alteration of these constants has only the effect of replacing Wi(a) or w2(a) by an equivalent pseudo valuation, and so, from the standpoint of valuation theory, it would suffice to put 5. Acta math. 41 (1919), 271284. 6. Acta math. 67 (1936), 5180.
VALUATIONS AND PSEUDOVALUATIONS
19
X = Xi = ... = Xj» = 1.
However, for the later applications to Diophantine approximations, a different choice of these constants is of advantage. For let g ** 2 be an arbitrary integer, and let
g = Pie'...pfr be its factorisation into a product of powers of different primes Pi ,•••> Pr with exponents ei ,..., er that are positive integers. Fix now Xi,..., Xr such that  g  ^ = . ... . . = l g £ r  J , i.e., take PI Pr Jog_g_ 1
logg e r logpr '
and put log g
log g
ag=max(a£ logpl ,..., lalj10"*)
and log g logpl
ag* = max(a, a^
log g ',..., aglogpr ) = max(a,ag).
We call ag,and ag* the gadic and the g*adic pseudovaluations, respectively, and also speak of the gadic and g*adic values of a. The definition of ag implies that for all a in F and all integers n. This is easily verified and is also contained in the property (II) of § 4. If g' ^ 2 is a second integer, it is obvious that ag and agf are equivalent if and only if g and gf have the same prime factors Pi,..., Pr and differ only in their exponents; and just the same holds for ag* and ag»*. Furthermore,
ag = ap if g is a positive integral power of the single prime p. In these lectures, P will denote the real field, i.e., the completiorkof P with respect to a; and similarly Pp. Pg, and Pg* will stand for the completions of F with respect to I alp, a[g, and ag*, respectively. Then Pp, Pg, and Pg* are the field ofpadic numbers, and the rings of gadic and g*adic numbers, respectively. This field and these two rings were introduced by K. Hensel7 ) in 1892 and have proved of fundamental importance in many branches of mathematics. * We shall study the elements of Pp, Pg, and Pg* in detail in the next 7. HensePs little book Zahlentheorie (Berlin 1913), which gives an elementary introduction to the theory of padic and gadic numbers, may be particularly recommanded on account of its many examples of actual computations with such numbers.
20
LECTURES ON DIOPHANTINE APPROXIMATIONS
chapter; and they will form both the object and a tool in most of the later work. It is clear that, while a and ag* are Archimedean, I alp and ag % are Non Archimedean. This was proved for ap in §4 and so follows for ag from the definition. It is sometimes convenient to define ag also in the excluded case when g = 1, by putting ai = w0(a). 15. Independent pseudovaluations. The gadic ring Pg and the g*adic ring Pg* can be decomposed into finitely many field P and Pp, as was already proved by Hens el7). We shall prove this decomposition as a special case of a more general theorem on ps eudo valuations . Denote by K a field, by wi (a),..., wr(a) finitely many valuations or pseudo valuations of K, and by 1 if h = k, 0 if h 4 k the wellknown Kronecker symbol. Then wi(a),..., wr(a) are said to be independent if there exists for each suffix h = 1, 2,..., r an infinite sequence
{da(hU(h);dlh),...} in K such that lim Wk(dj? 6hk) = 0
(k = l,2,...,r),
ejik = lim djj (wk) m —>°°
(k = l,2,...,r).
or, what is the same, that
From this definition, it is immediately clear that, if wi(a)~ wi(a),..., w' r (a)~ wr(a), then also wi(a),..., wr(a) are independent. By way of example, let us consider the rational field F. Here the following result holds. (1):
y ?!»•••> Pr are finitely many distinct primes, then the valuations lal, lapl,..., lalpr and hence also the valuations apl,..., apr are independent. First, the sequence m \v>\ l m >, whprp where drfm  (plpa ...— Pr)
VALUATIONS AND PSEUDO VALUATIONS
21
is easily seen to have the limits 1 with respect to I a I and 0 with respect to apl,..., apr, respectively. Secondly, select for each suffix h = 1, 2, ...,r a positive integer a^ such that Phh > PiPa.PrThen it is not difficult to verify that the sequence />)} where d fr) (piP2...Pr)m idm J , where dm/  ( Pl p 2 ...p r )m +p a h m » has the limit 1 with respect to lalp^, but is a null sequence with respect to I a , as well as with respect to all I a pk where k 4 h. Kgi^2,..., gr ^ 2 are finitely many integers which are relatively prime in pairs, one shows by a similar proof that also a, agl,..., agr and hence also agl,..., agr are independent. 16. The decomposition theorem.
Let again K be a field, and let wi(a) ,..., wr(a), where r ^ 2, be finitely many independent valuations or pseudo valuations of K. As in § 4, we put ws(a) = max[wi(a),..., w r (a)] and further write wjtya) = max wk(a) (h = l,2,...,r). These functions are likewise valuations or pseudovaluations of K, and w«(a) is, what we call the sum of Wi (a),..., wr(a). The following lemma gives the justification for the term of "independent" valuations or pseudo valuations. (m): Let ^eK^,..., areKWp be arbitrary elements of the completions of K with respect to Wi (a),..., Wr(a), respectively. Then there exists an infinite sequence {am} in K such that, simultaneously, lim am = a n (wh)
^h = 1 > 2 v> r )«
First, ai,..., ar are defined as the limits lim a } ( w h ) m—»°°
(h =
of certain infinite sequences {aW},..., {a^ty in K. Secondly, by the definition of independence, there also exist r infinite sequences {dm)},..., {dW}
22
LECTURES ON DIOPHANTINE APPROXIMATIONS
in K satisfying )
(h,k = l,2,...,r),
In particular, each such sequence {dmT» is a null sequence with respect to all wk(a) where k ± h, and so it is a null sequence also with respect to
w(a).
There exists then, for each h, an infinite sequence of strictly increasing suffixes mfci, mh2> mh3>— such that
Thus
and hence, from the definition of wyO (a), (A):
Ihn d^
ajh) = 0 (wk)
(h,k = l,2,...,r; h*k).
On the other hand8),
? = 1 (wh) (h so that (B):
^m d
a!
= Urn d
£m a
= 1. «h = «h(wh)
(h = l,2,...,r).
On combining (A) with (B), it follows that the new sequence {am} where ai= Z dmhiaih) h=l
(1 = 1,2,3,...)
has the required limits a\ ,..., ar with respect to Wi (a),..., wr(a), 'respectively. We finally prove the following decomposition theorem which establishes the connection between the completions KW, KWl,..., KWr of K. (n):
There is a onetoone correspondence
8. Letfbuj} be a fundamental sequence with respect to w(a), and let (bm]}i where nii < m2 <ms < . . ., be an infinite subsequence offb^. Then {to bm , b2b bsbm » . . .} evidently is a null sequence with respect to w(a), whence lim bm = lim bmi(w) . m—»oo l—> oo
VALUATIONS AND PS EUDO VALUATIONS
23
between the elements a of Kws and the ordered sets («i ,..., «r) of one element in each of KWl ,..., KWr such that, if a— (ai,,.., ar)
and /3 —(ft,..., /3r),
$&en also correspondence is defined by lim am(ws),
a^ = lim a m (wh)
(h = l,2,...,r),
{am} te a sequence in K wfo'cfe is a fundamental sequence with respect to all of wi (a),..., wr(a), and ws(a). • First, we note that it is obvious from the definition of w^(a) that a sequence {am} in K which is a fundamental sequence with respect to each of Wi(a),..., wr(a) is also a fundamental sequence with respect to ws(a), and vice versa; similarly, if the sequence is a null sequence with respect to each of Wi(a),..., Wr(a), then it is also one with respect to ws(a), and vice versa. Now, by lemma (m), the arbitrary elements aj.eKWl ,..., areKWr can be defined as limits an = lim am(wh) m— » °° of the same sequence {am} in K, and then the limit
(h = 1,2,. ..,r)
a= lim am(w;s)
defines an element a of Kw^. Conversely, if a is given as the limit with respect to w^(a) of such a sequence .{am}> then the limits of {am} with respect to wi(a),..., Wr(a) likewise exist and define elements ai,..., ar of KWl,..., KWr, respectively. The relation a *^(ai ,..., ar) is independent of the special sequence {am} used in the definition of a, ai9..., ar. For if (C): lim am= lim am(wn) (h = l,2,...,r), m—*°° m—»°o then {amam} is a null sequence with respect to each of wi(a),..., wr(a) and hence also with respect to w^(a); therefore (D):
lim am = Um a m—»«)
Conversely, (D) implies again (C). The formulae for a+0, a0, and a/3 finally follow at once from the rules for limits proved in § 9. If a*^(ai,..., ar), oJi,..., ar are called the components of a. All the completions KWl,..., KWr, and KWS are extensions of K, and every element a of K lies simultaneously in each of these r+1 rings or fields. For such and only for such elements the correspondence relation takes
24
LECTURES ON DIOPHANTINE APPROXIMATIONS
the simple form a ~^""^" \a, a,..., a j. In particular, 0~ 0, there is a positive number q(e) such that
(
m n v £ ak  £ at) = w(an+i + an+2 + ... + a m )< e k=l k=l /
for all integers m, n satisfying m > n ^ q(e). Since w(an+i + an+2 + ... + am) * w(an+l) + w(an+2) + ... + w(am), 00
the series
^
a
m certainly converges if the series of real numbers oo
E w(am) m=l
converges; but the converse need not even be true in real analysis where w(a)=a. On taking m=n+l, it is also obvious that Y am cannot be convergent m=l unless lim am = 0 (w); m—>°o
but, just as in real analysis, this condition is not in general sufficient for convergence. There is, however, one important case when it is sufficient, viz. that when w(a) is NonArchimedean. For now w(an+l + an+2 + ... + am) ^ max[w(an+l),w(an+2),., w(am)]. and so, if {am} is & null sequence with respect to w(a), the righthand side is smaller than e f or all m > n In the following chapter these simple remarks on convergent series will be applied to series for padic, gadic, and g*adic numbers. Final remark: The sketch of valuation theory given in this chapter has been strictly limited to those facts that are to be applied later. For further study of this interesting and important theory the following texts may be referred to: E. Art in, Algebraic numbers and algebraic functions I, Princeton 1951. H. Hasse, Zahlentheorie, Berlin 1949. O. F. G. Schilling, Theory of valuations, Math. Surveys IV, Amer. Math. Soc. 1950. H. Weyl, Algebraic theory of numbers, Princeton 1940.
Chapter 2 THE pADIC, gADIC, AND g*ADIC SERIES Historically, K. Hensel was led to his padic and gadic numbers by considerations of analogy to function fields. Let S be the complex number field, x an indeterminate, and K = S (x) a simple transcendental extension of S; let further w(a) be any valuation or pseudovaluation of K with the property C, i.e., such that w(c) = w0(c) if ceS, where Wo(a) denotes the trivial valuation defined in §1 of Chapter 1. It can be proved that every valuation with the property C must be equivalent to one of the valuations wo(a), llall, apr introduced in § 3 of Chapter 1; however, now every "prime" p has the special form p=xc where ceS because S is algebraically closed. One can further show that every pseudovaluation with the property C either is equivalent to one of these valuations, or it is equivalent to a pseudovaluation of one of the two forms wi(a) = max(apl,..., apr) and wa(a) = max(a, lapjl,...,apr). Here Pi = x  ci,..., pr = x  cr, where ch =(= % if h 4 k, are finitely many distinct "primes", and we have r ^2 in the case of Wi(a) and r ^ 1 in that of wa(a). The position is thus analogous to that mentioned in § 14 of Chapter 1 for the rational field T , with Hall, aL, w^a), W2(a) corresponding to I a I, I a I p, I a I g, I a I g*, respectively, There is, however, the difference that all these valuations and pseudovaluations of K are NonArchimedean. It is not difficult to prove that the completion of K with respect to l l a l l is the field of all formal series
while that of K with respect to I la Up, where p=xc, is the field of aM formal series cf(xc)f + cf+i(xc)f+1 + Cf+2(xc)f+2 + ... . In both cases f may be any rational integer, and the coefficients cm may be arbitrary elements of S. The convergence of the series follows from the results in § 17 of Chapter 1 because H i l l  e< 1, cmll< 1, and xcL=0 < 1, c m L«l f 26
THE pADIC, gADIC, AND g*ADIC SERIES
27
respectively, and hence lim c m (M  = 0, lim lcm(xc)mp = 0. \x/ m—»°° * In both cases the constant field S has the algebraic property of being the residue class field K/x and K/xc, respectively. Similar, but slightly more complicated developments hold also for the completions of K with respect to Wi(a) and W2(a), but there is no need to go into details. Consider now the valuation I alp of F and the corresponding padic completion Pp of F. We have m_,00
I P I p =  < !> and c L ^ 1 for all rational integers c. It follows that every formal series Cf p* + cf+lp*"1"* + cf+2P*+2 + ••• where f and all the coefficients Cm are rational integers, converges with respect to ap; for the valuation Tap is NonArchimedean, and It will be proved in this chapter that every element of Pp can be written in many ways as a series of this kind, but that there is one and only one series in which the coefficients assume only values in the finite set {0,1,..., pl}. When Hensel discovered the padic numbers towards the end of last century, there was not yet any general field theory or theory of valuations. He defined his numbers by the series and by the rules for computing with them. In this work he followed the analogy to the Laurent series /l\f
fiY+1
f!V+2
or cf (xc)f + cf+i(xc)f+1 + cf+2(x2)f+2 + ... for an analytic function in the neighbourhood of a pole, either at x=°° or at a finite point x=c. Such series are convergent in the sense of complex analysis rather than with respect to the valuations a or I la Up; but even in function theory the latter kind of convergence plays a big role in connection with the orders of poles and zeros. The investigations of this chapter are concerned only with the padic, gadic, and g*adic numbers. However, the method is much more general, and it can in particular be used to prove the earlier assertions about the completions of K with respect to a and I la Up. 1. Notation.
In this and the later chapters the notation will be essentially the same as before. Always PI,..., pr denote finitely many distinct primes, and g ^2 denotes an integer with the factorisation
28
LECTURES ON DIOPHANTINE APPROXIMATIONS g = piei... prer
where ei,..., er are positive integers. The valuations la I and I a p, and the pseudo valuations ag and ag*, are defined as in Chapter 1, and P, Pp, Pg, and Pg* denote the corresponding completions of the rational field F, thus are the fields of the real and the padic numbers, and the rings of the gadic and the g*adic numbers, respectively. We shall in general use Latin letters for rational numbers, small Greek letters for real and padic numbers, and capital Greek letters for gadic and g*adic numbers. If A^**~(cti ,..., otr) is a gadic number with the pjadic components a] for j=l, 2,..., r, the gadic value of A is equal to logg logg
Similarly, if A*~~(a, a lf ..., ar) is a g*adic number with the real component a and the opadic components «j for j=l, 2,..., r, the g*adic value of A* is given by
(R
logg laiig10™1 ,..., i
J ^J These two limit formulae imply that lim l a m  g =  A  g
and
(j = 1,2,...,r). ' ' '
lim ampj = l«jpj
(j = l,2,...,r).
Now, by definition,
(
log g
lamlpi
10gPl
logg ,, la m lp?
10gPr
\ / ,
and so the assertion follows immediately. We note that (I): Ung = (Ug)n, U*ng* = (U*g*)n for all AePg and A*ePg* and aU positive integers n; and that further (II): Ug n lg= for all AePg and all rational integers n. These properties follow easily from the explicit expressions for Ulg and U*g*, and from lglg = g"1, lgl pl =Pi"ei,..., l g l p r = P r e r .
THE pADIC, gADIC, AND g*ADIC SERIES
29
They are special cases of the properties (I) and (II) in § 4 of Chapter 1. Here and later, we have continuously to deal with limits
Urn ... (w) m—»°° where w(a) stands for one of a, ap,
ag, or ag*.
In order to shorten the formulae, we shall always omit the sign a of the absolute value and write
lim ... m—»°° when dealing with real limits. We further shall replace
and
lim ... (ap) m—x»
by
lim
by
... (ag)
lim ... (aL*) m— »°°
by
lim ... (p), m— >«> lim ... (g),
m— — >o
lim ... (g*), m— >, U(m>  AM L « ± . h & g
It follows that, for all m 5* f,
= A« Hf
and
34
LECTURES ON DIOPHANTINE APPROXIMATIONS Here
~
I •» *
g
' whence
lim Therefore A' ' can be written as the convergent infinite series X® = A» + f A y(a, mjA^a). The proofs of both theorems depend very essentially on the Fundamental Inequality. The other two general properties mentioned in the introduction have not been used. They form the basis for the next investigations. 5. A theorem on linear forms.
From now on the number a need no longer be algebraic. Our aim is to find polynomials F(x) for which o>{F(a)} is small. The construction makes use of Dirichlet's principle and of the density properties of the integers which were mentioned in the introduction. In the special case when F(x) is a linear polynomial, a simpler and more explicit method will be given in the * next chapter. We begin'with a general theorem on linear forms1 Theorem 2: Let
n Lh(x) = u ahkxk (h = 1,2,..., n) k=l be n linear forms in n variables with real coefficients, and let n a= max £ lahkl>0h=l,2,...,n k=i
If Xi,..., Xn are n positive numbers such that Xi...X n > a11, there exist n integers xi,..., xn not all zero satisfying L h (x)<X n
(h = 1,2,..., n).
1. This is a slightly weakened form of Minkowski's theorem on linear forms (Geometrie der Zahlen, §§ 3637). I learned the proof as given here more than 30 years ago from my teacher C. L. Siegel.
A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS
49
Proof: Denote by N a very large positive integer, and define n further positive integers ti,..., tn by the formulae
t h LTAhT J
+1
(h = 1,2,..., n).
Since, asymptotically, ti... tn  . a" (2N+l)n AI .. .AH
as N » oo,
N can be chosen so large that ti...t n < (2N+l)n. The ordered system (xi,..., xn) is said to be admissible if each xh equals one of the 2N+1 integers 0,71,72,..., 7N between N and +N. There are then (2N+l)n admissible systems. For such systems, n n Ln(x) = I E a h k x k l ^ N E l a f c k l ^ a N , k=l k=l
so that aN ^ Lh(x) ^ +aN.
Divide the interval [aN, +aN], for each suffix h=l, 2,..., n, into th subintervals of equal length y^, the subintervals j[h\ J^\...9 j£', say; points on the boundary of two adjacent intervals should be counted as belonging to only one of them. In accordance with the values assumed by the forms Li(x),..., Ln(x), there corresponds to each admissible system (xi,..., xn) (1) (n) (h) a unique ordered system (Jj ,..., Jj[n ) of n subintervals such that Lh(x)eJ£ . By the construction, the number of all such systems of n subintervals is exactly ti,... tn, hence is less than the number of admissible systems (xi,...,xn). It follows then, from Dirichlet's principle, that there exist two distinct admissible systems (xi,..., xn) and (xi',..., xtf) for which the values of Li (x'),..., Ln(x') and of Li(x"),..., Ln(x") lie in the same system of subintervals (Ji7,..., J{ L A 0 Therefore l n (h = 1,2,..., n). *•»!
Put f
tf
T
.
It
Xi = Xi  Xi ,..., Xn = X!n  Xn.
Then xi,..., xn are integers not all zero for which Lh(x)= Lh(x!)  Lh(x")«s T— —
6. On a system of both real and padic linear forms. From Theorem 2 we shall now deduce a result on the values of systems of linear forms that have coefficients in different completions of the rational field. Denote by a linear form with real coefficients not all zero for which further by pi ,..., pr finitely many distinct primes, and by lj(x) = «j1x1+...+«jMxM a linear form with pjadic coefficients satisfying
(] = 1,2,..., r)
maxdaj! lpj,..., I«JM!PJ) * 1 (J = M,.,., *)• Since l(x) has at least one nonzero coefficient, there is no loss of generality in assuming that aM* 0;
for, if necessary, it suffices to renumber the variables. Let Ei,..., Er be r arbitrary positive integers. By hypothesis, the coefficients aj^ of each form lj(x) are p^adic integers. Hence there exist rational integers aj ^ satisfying the inequalities
Now put
ajLx1+ ... + ajM XM +
Lj(x)
pf 1 xM+j}
(j= 1,2,..., r)
if j = 1,2,..., r, if j = r+pi, /!= 1,2,..., Ml, if j = M+r
X][jl
l(x)
The linear forms Lj(x) so defined have real coefficients such that the sum of the absolute values of the coefficients of each form is not greater than 1. Therefore, on applying Theorem 2, with n=M+r and a=l, it follows that there exist integers xi , x2 ,..., XM+r n°t oil zero satisfying the system of inequalities Lj(x) < Aj
(j = l,2,...,M+r),
A TEST FOR ALGEBRAIC OR TRANSCENDENTAL NUMBERS whenever Xi , Xa ,..., AM+r
are
51
positive numbers such that
Let us now specialize this result in two different ways, First specialization: Denote by T a number greater than 1 and define a number t > 1 by tMl further put if j = 1,2,..., r, i f j p+ttji.l,2,...,Ml,
*J 4
HJ
Then and so we may apply the last result. It follows then from the formulae for Lj(x) and Xj that there exist integers xi,x2,..., XM+r not all zero satisfying the inequalities llfto+pfJxM+jk I (j = 1,2,..., r), Ixj< t
( M = 1,2,..., Ml),
The first r of these formulae imply that already xi, ..., XM cannot all vanish; for otherwise also XM+I = ... = XM+I. = 0. These first r inequalities are equivalent to
because the expressions on the lefthand sides are integers. Hence > = 0(modpj E J)
(j = 1,2,..., r)
and therefore ll](x)lpj« P ]  E J
(j = 1,2,..., r).
Now M
= I E so that also
(j = 1,2,..., r).
52
LECTURES ON DIOPHANTINE APPROXIMATIONS
Finally, from T >1 and t > 1 and from the last M inequalities, we deduce that
Since <XM! * 1, it follows that nx
/I I I \\ ^ •• max(xi I,..., IxMlX i *» aMi •
Second specialisation: Put Kx)=x M which is evidently allowed, and choose T=J
where tM = 2(M+l)r p?1... pEr .
Otherwise leave the notation just as in the first specialisation. On repeating the last computations, it now follows that there exist integers xi,..., XM not all zero such that lj(x)pj< P : E J a = 1,2,..., r), maxdxil,..., Ixjvjl) < t. The two results just proved contain the following theorem. Theorem 3: Let l(x) = «ixi+...+ «MxM, where 0< \a\ I+...+«M < 1, be a linear form with real coefficients; let PI ,..., pr be finitely many distinct primes; let lj(x) = ofjlXl+...+ ajMxM, where max(aj1lp],...,ajMpj) < 1, be, for j = 1,2,..., r, a linear form with pjadic coefficients; let E!,...,Er be positive integers; and let T> 1. (i): There exists a positive constant Ai independent of Ei,..., Er, and T, such that there are integers xi,..., XM for which Ei it / \i — ~E. px ^ PI ,•••> H r W l p r ^ P r »
0
{F(a)} « y (a,m)u" fl( * m) , 0 < A ^ u, because F(a) * 0 by the hypothesis. Assume further that the parameter u tends to infinity, so that co{F(a) } tends to zero. This implies that F(x) cannot remain fixed, but must run over an infinite sequence of distinct polynomials with heights A tending to infinity. For there are only finitely many polynomials F(x) with integral. coefficients, of degrees not greater than m and of bounded heights; and for such a set of polynomials co{F(a)} necessarily has a positive minimum. Hence, if a is transcendental, there exist infinitely many distinct
56
LECTURES ON DIOPHANTINE APPROXIMATIONS
polynomials F(x) with integral coefficients, of degrees not exceeding m and of heights tending to infinity, such that 0 < o>{F(a)} This is true whatever the value of m. We may then choose m so large that jjt(d,m) is greater than any prescribed positive number A and that therefore as soon as A is sufficiently large. This result may be combined with Theorem 1, leading to the following test for transcendency. Theorem 5: The number a (which may be real, padic, gadic, or g*adic) is transcendental if and only if, given any positive number A, there exist a positive integer m and infinitely many distinct polynomials F(x) with integral coefficients, of degrees not exceeding m and of heights A tending to infinity, such that 0 < o>{F(a)} ^ AA. This test leads immediately to a special class of transcendental numbers, the Liouville numbers2 . A number a is said to be a Liouville number if there exist, (i) an infinite sequence of distinct rational numbers l, Hn = max(Pn,Qn), and (ii) an infinite sequence of positive numbers {Ai, A2, A3,...} tending to infinity, such that 0o], «i > 1, 1
Znr Approximation algebraischer Zahlen III, (1934), Acta math. 62, 91166. See, in particular, the first part of this paper. 58
CONTINUED FRACTIONS
59
on = ai + —, where ai = [cti ], aa > 1, «2
•
anl = a n _i + —, where a n _i = [anlL «n > 1. For shortness, we express this set of n formulae in the abbreviated form «o = [ao,ai,..., anl, «n] which stands for the explicit formula 1 = ao +
a2
obtained by eliminating HI, Q!2,..., anlIf the algorithm terminates with the integer an, then Qfn = an and ao = [ao, ai ,..., an»i, an]. We call the symbol on the righthand side a, finite continued fraction for a0. If, however, the algorithm never breaks off, then we write ao = [ao,ai, fe ,...], and say that the symbol on the righthand side is an infinite continued fraction for a0. For the present such an infinite continued fraction simply expresses the fact that ao, ai, a2,... are the successive "incomplete denominators" of o?o as given by the algorithm. We note that if the continued fraction for a0 is finite and ends with an, where n ^ 1, then an ^2 because an > 1. 2. The convergents of the continued fraction for a0.
Assume that either «0 = [ao, ai,..., a n _i, an] or that a0 = [ao,ai, a*,...]. Then define integers Pfc and Qk by the formulae (P! = 1)
(Po = ao)
jPk = a k P k _i + Pk.2)
(01 = 0)'
(Qo = i) ' (Q k =a kQki+Qk2)
ifk
^lj
where k is not greater than n in the first case, but is unrestricted in the second case. p The rational numbers ^ are called the convergents of a0. They are already written as simplified fractions,
(Pk, Qk) = 1,
60
LECTURES ON DIOPHANTINE APPROXIMATIONS
because (1):
PfclQk  PfcQk1 = (Dk if k ^ 0. This equation trivially holds f or k = 0 because
PiQo PoQi = +l. Assume next that k^l and that (1) has already been proved for the suffix kl. Then PfclQk  pkQkl = pkl(akQkl + Qk2)  (afcPfc1 + Pk2)Qkl = =  (Pk2Qkl  PklQk2) =  (I)15"1 = (l)k, proving the assertion also for the suffix k and so generally. Next, oto may be written as + Pk2
This formula is certainly true for k=l because 1 a 0 ai+l « 0  30 + — = —
—
Assume further that k ^ 2 and that it is already known that _ Pk2akl + Pk3 °~Qk2akl+Qk3 ' Since a^l = afcl+ — , it follows then that
«o 
Pk2
.
1 +
j^
+ Pk3

Pk,2
Pkl«k + Pk2 9
Qk2 (ak1 +—J +Qk3 (aklQk2 + Qk3)«k + Qk2 Qkl^k + Qk2 giving the assertion also for the suffix k and so generally. From (1) and (2) it follows in particular that (3)
a  Pk"1 = (I)1*"1 ° " Qk1 " Qkl(Qkl«k + Qk2) '
because p kl ( p kl a k +p k2)Qkl p kl(Qkl af k+Qk2) pk2QklQklQk2 a0  ——= = . Qk s ' Qkl(Qkl«k + Qk2) Qkl(Qkl«k + Qk2) 3. The distinction between rational and irrational numbers. It can now be shown that the continued fractions of rational numbers are finite, those of irrational numbers are infinite. First, every finite continued fraction o?o = [ao, ai,..., 3nljan] has a rational value. For, by (2), applied with k=n and «n=an>
CONTINUED FRACTIONS
61
' 1'""> n" ' nJ Qnlan + Qn2 Qn " Conversely, if aQ is a rational number, the continued fraction algorithm for a0 breaks off after finitely many steps. For let the trivial case a0=ao be excluded, and let ao,ai,a2,... and 0i,a2,... be defined as in §1. Then all numbers «k are rational, say — where (pk, q^) = 1 and qk ^ 1. Further ril = ak1 K +—,
here fakl> Pklakl 0, Pk1  akiqk1 = Qkl(«kl  afcl) ="^" j< q^!
It follows that Pk = Qk1,
Ok = Pk1 " akiqk_i < qfc1,
and that therefore qo > qi > ... > 1 . There is then a finite suffix k=n such that Qn=l> and the algorithm terminates with an=an. The result so proved implies that irrational numbers always have infinite continued fractions OiQ = [80,81,81,...].
This continued fraction converges to oto in the sense that (S):
a0 = [ao,ai,a2,...] = lim ^
K—»oo ^K
= lim [ao,aa,...,akl,ak]. K— *°°
To prove this assertion, we first note that, by the definition of ak, ak ^ «k < afc + 1,
hence that Qk = Qkiak+Qk2 * Qkiafc^Qk2 < Qkl(ak+D+Qk2 = Qk+Qk1 ^ 2Qk . Therefore, from (3), teV '
1 < 1 ^ )„ Pk1 1 < 1 2QklQk " Qkl(Qk«kl) 1° " Qk1 1 " QklQk ' (This formula remains valid for rational ot0 provided that k < n.) W>
62
LECTURES ON DIOPHANTINE APPROXIMATIONS Now
Qo = 1, Qi = ai ^ 1, and Qk = akQfcl + Qk2 > Qk1 + 1 if k > 2, so that It follows then from (6), for irrational numbers a0, that 0
as k*oo,
as was asserted. In fact, even the stronger relation Urn (Qk«oPk) = 0 holds because i_
i.l
.1
as
4. Inequalities for Qk lQki<Xo  Pfcil+ lQk«oPkl> This gives the following result. V (P, q) = 1 and 1 ^ q ^ Qk, then q«o  pl with equality only if p = Pfc and q = Qk.
The convergents of a0 are thus, in a very strong sense, its best approximations. 6. The rational approximations of gadic integers.
After this short sketch of the basic properties of continued fractions for real numbers, we proceed to the study of the continued fractions for gadic and g*adic numbers. There is no need for dealing separately with the case
64
LECTURES ON DIOPHANTINE APPROXIMATIONS
of padic numbers because these may be considered as special cases of gadic numbers. We begin with the study of gadic numbers, but, for simplicity, consider only gadic integers A * 0; thus As was proved in §5 of Chapter 2, such gadic integers may be defined explicitly in terms of gadic series
... (g) 0
2
where the coefficients A* ) A^), A< ),... are integers 0, 1, 2,..., g1. Put Am = A(°) + A«g + A(2)g* + ... + AtoDgml
(m = 1,2,3,...
so that Am is a rational integer satisfying 0 V6*. Thus now the second case of §7 holds, and there are no solutions of both (16) and (17). However, we have now the solution m a
of (18).
m
, 0 < 2 < 62 ,
are
CONTINUED FRACTIONS
69
9. Final remarks to the gadic algorithm. I believe that the algorithm sketched in the last sections is worthy of a more detailed study, and I have little doubt that many interesting properties will then be discovered. One possible approach arises from the following facts. The numbers ,(m) Am ° g51 occurring in the algorithm are not independent. Since
^
is connected with ct^'
by the relation a 0 ( m ) +A ( m )
m
Here A may assume only the g values 0, 1, 2,..., g1. There is thus associated with A an infinite sequence of linear transformations {Ti, T2, TS,...} where g This sequence is thus formed by repeating not more than g distinct elements. WJien A is rational, the sequence is periodic; i.e., there are two positive integers m0 and n such that T
m+n = Tm if m ^ m0. One may therefore expect some simple laws relating to one another the continued fractions of consecutive numbers oto and ao + . It further seems probably that there is some nontrivial connection to the theory of the modular group and its congruence subgroups. In a very similar theory for padic numbers this was indeed the case as I proved in an earlier paper2. There would be no difficulty in extending the method of that paper to the gadic case. 10. The continued fraction algorithm for g"adic numbers.
The continued fraction algorithm for gadic numbers has an analogue for g*adic numbers. We shall consider only such g*adic numbers
2
Annals of Math. 41 (1940), 856.
70
LECTURES ON DIOPHANTINE APPROXIMATIONS
which have the property that their gadic component A is a gadic integer; for the real component a no restriction is necessary. The component A may again be written as a gadic series •< A = A(°) + A( 1 )g + A(2)g3 + ... (g), where the coefficients A(m) are integers 0, 1, 2,..., g1. Just as in §6 put Am = A(°) + A«g + ... + A(m1>gm1, so that
U  Amg < gm, 0 < Am ^ gm 1. We found then that the integral solutions P, Q of the inequality are identical with the integral solution P, Q of
QAm  P = gmR where R is a further integer. Assume now that such a solution P, Q has the additional property that p Q is a close approximation to the real component a. Then
QaP = Qa  (AmQ  gmR) = Q(a  Am) + gmR is small. We therefore put (m)_Ama 0° = gm and demand that
•p
(m}
is close to 0o . This leads to the following algorithm.
Develop the real number 3o
into a continued fraction (m) r (m) (m) (m) , 0o = [bo , bi , ba ,...J.
,Here bo ba
is an integer which may be positive, negative, or zero, and bi
,
, bs
,... are positive integers. The continued fraction terminates if / \ and only if a and hence also 0o is rational. R.(m) As in the gadic case, denote by \ the convergents of this continued
fraction so that
I?' = 0
Q<m) = l
Q(m) k
for k = 1,2,3,...
Further put again = AmQ(km)gB>R(km)
(k = 1,0,1,...).
CONTINUED FRACTIONS
71
Then, as before, (P[m), Q^) to « divisor of gm because (Q(km), R(km)) = 1. The construction implies that
lQ ( k^P ( k m) lg*g m . Further, by the equation (3), (m) 0 (m)/ 0Q Q
(m) o(m kl < klP k + Q
where j3k is the real number analogous to the former number o?k that belongs to the continued fraction. This equation may be written as
In this equation,
.(m) . Jm). , (m) so that, similarly as before, (m) n(m) . n(m) n(m) n (m) n(m) Qk ^ Qk1 ^k + Qk2 < Qk + Qkl ^ Hence, on changing from k to k+1, it follows that
9n(m) 2Q
k
•
Exclude the case when a is rational, so that this inequality is valid for all k?0. Since there exists then for every positive integer m a suffix k such that (m) . 2m fAm) Qk < g * Qk+1 » and for this suffix, both Q[mWkm)* gm and Q ( k m) AP ( k m)  g ^ Now the g*adic value of any g*adic number J3*— (]8,5) was defined by the equation
gm.
72
LECTURES ON DIOPHANTINE APPROXIMATIONS
The following result has thus been obtained. Let the real component of A*~+*~(a,A) be irrational, and let the gadic component be a gadic integer. For every positive integer m, the continued fraction algorithm allows to construct a pair of integers P f c l ) . Q k a ) > 0 such that QJf >A*  P(km)g* * gm, 0 < max(P(km), Q^) * g2m. This result remains true for rational a, as can be shown, but it then takes a rather trivial form. For now
as soon as m is sufficiently large. The remarks made with regard to the gadic algorithm in §9 may be repeated for the g*adic algorithm. For also here consecutive numbers j8o ' and j3o + are again related by the transformation Tm.
PART 2 RATIONAL APPROXIMATIONS OF ALGEBRAIC NUMBERS The problem and its history. Let a be a real algebraic number of degree n^2; thus a. is irrational. One of the results obtained in the proof of Theorem 1 of Chapter 3 was as follows. Let F(x) = Aoxm + Aix™1 + ... + Am * 0 be any polynomial with integral coefficients, of degree at most m, and of height A = F(x)=max(lAol, Aj,..., A m ) £ 1. Then either F(a) =0 or F(a) 1 5* Ci (m)^m'l\ where ci(m) > 0 depends on a and on m, but not on A. Let now m=l and F(x)=Qx'P where Q > 0 and P are integers; then A=max(fp, Q), and on putting GI = Ci(l), the last result implies that because QaP* 0. This inequality is equivalent to (1):
where c > 0 is another constant depending only on a . For either and then
Q,n
or
^2 the inequality (3) has at most finitely many rational solutions and hence that there is then a constant y (a,p) > 0 such that PI P aQ Py(a,p)Q~P for all rational numbers Q .
I
However, no method is known for actually finding such a constant y (a,p), a disadvantage shared by the methods of Thue, Siegel, Dyson, and Roth, and also by the work in this second part.
2. Skrifter udgivne af VidenskabsSelskabet i Christiania, 1908, and J. reine angew. Math. 135 (1909), 284305. 3. Math. Z. 10 (1921), 173213. 4. Acta math. 79 (1947), 225240. 5. Mathematika 2 (1955), 120 and 168.
APPROXIMATIONS OF ALGEBRAIC NUMBERS
75
Before Roth, Th. Schneider6 had already proved a weaker theorem. Assume there exists an infinite sequence of rational numbers
!?•§•'•!•' •••' where i * QI < ^ < Qs < •••'
such that (4):
a^*Q^
(k = 1,2,3,...)
for some p >2. Then
This theorem by Schneider is nearly as powerful as Roth's theorem for certain applications to proofs of transcendency. Generalisations of these results by Siegel and Schneider have been known for many years. Already Siegel himself8 extended his result to the approximations of a by the numbers of an arbitrary algebraic number field of finite degree. The corresponding analogue of Roth's theorem has recently been established by W. J. LeVeque7 . As these lectures do not deal with the padic completions of algebraic number fields, such generalisations will not be discussed. But it would have much interest to carry out a similar extension of the later Approximation Theorems. See, however, Appendix C. In another kind of generalisation, the numerator P or the denominator Q p of the rational approximation •= is restricted by some arithmetic condition. For instance, it may be demanded that Q is a power of a given positive integer or that, more generally, the greatest prime factor of Q is bounded. Such theorems were given by Schneider6 and myself8, but asserted only a result of the form (5). However, now that Roth's method is known, D. Ridout9 has obtained an extension of this kind for Roth's theorem which is free of this defect. A third kind of generalisation will seem natural to the reader of the first part. Instead of studying the rational approximations of a real algebraic number, one considers those of a padic, gadic, or g*adic algebraic number a. In the notation of Chapter 3, it is then especially the behaviour of the function
which is of interest. Some 25 years ago, I10 studied exactly this kind of problem by means of Siegel's methods. Again Ridout11 has obtained the analogous extension of Roth's theorem. 6. J. reine angew. Math. 175 (1936), 182192. 7. Topics in number theory, vol. 2, chapter 4 (Reading, Mass. 1956). 8. Proc. Kon. Akad. Amsterdam 39 (1936), 633644, 729737; Acta Arithmetica 3 (1938), 8993. 9. Mathematika 4 (1957), 125131. 10. Math. Ann. 107 (1933), 691730; 108 (1933, 3755). My results have been extended to the approximations of padic algebraic numbers by C. J. Parry, Acta math. 83 (1950), 1100. 11. Mathematika 5 (1958), 4048.
76
LECTURES ON DIOPHANTINE APPROXIMATIONS
The aim of the following chapters may now be stated as follows. We shall combine the method of Roth with the idea of Schneider on arithmetic restrictions for P and Q and that of mine on the use of padic algebraic numbers. By deliberately applying gadic numbers and the gadic pseudovaluation, it will be possible to simplify many of the proofs, as compared with my old paper10. Although the following proofs will make essential use of both real and gadic numbers, at least one form of the final results will be completely free of these numbers and state a property of rational numbers only. Thus real and gadic numbers will serve as tools, but not as an end in themselves. This seems to me highly satisfactory. For the theory of numbers still has its main interest in what it can tell us about the rational numbers and the rational integers. But if we want to find properties of the rational numbers, nothing must stop us in the choice of methods used for this purpose.
Chapter 5 ROTH'S LEMMA 1. Introduction Roth bases the proof of his theorem on a general property of polynomials which is to be proved in this chapter. This property is roughly as follows. Let A( Xl ,...,x m )=
2 ii=0
im=0
be a polynomial in m variables, with integral coefficients which are not "too large" in absolute values. Assume that
is a "very small" positive number. Further let
_ PI Qi ' *"'
Pm Qm
be m rational numbers written in their simplified forms for which both the maxima Hi =max(Pi,Qi),..., H m = max(P m l, iQml) and the quotients logH2 logH3 logHm log Hi' logH2''"' logHm1 are "very large". Then A(XI,..., xm) cannot vanish to a "very high" order at xi = KI ,..., xm=Km. (An exact formulation of Roth's Lemma will be given at the end of this chapter). The main idea of the proof consists in an induction for m, the number of variables, the case m=l being trivial. This induction uses a test for linear independence of polynomials in terms of the socalled generalised Wronski determinants. 2. Linear dependence and independence. Let
fy = fj/(xi,..., xm)
(v = 1,2,..., n)
be n rational functions of m variables, with coefficients in a field K. The functions are said to be linearly dependent (or for short, dependent) over K if there are elements ci,..., cn of K not all zero such that 77
78
LECTURES ON DIOPHANTINE APPROXIMATIONS
Cifi+... + cnfn 5 0 identically in Xi ,,.., xm. If no such elements exist, then the functions are called linearly independent (or for short, independent) over K. Evidently, if fi ,..., fn are independent, none of these junctions can vanish identically. Assume, in particular, that the coefficients of fi ,..., fn lie in the rational field F , and that these functions are dependent over the real field P. Then the functions are also dependent over r. For the identity cifi +...+cnfn = 0 is equivalent to a finite system of linear equations ci^ior + ... + cntoia = 0 (a = 1,2,..., s) for GI,..., cn with rational coefficients $va. By the hypothesis the rank of the matrix of this system of equations is smaller than n. The system has therefore also a solution ci,..., cn in rational numbers not all zero, whence the assertion. Conversely, if fi,..., fn have rational coefficients and are independent over F, then they are also independent over P. 3. Generalised Wronski determinants. The letter D, with or without suffixes, will be used to denote differential operators of the form
8Ji + ... + Jm where ]i,..., jm are nonnegative integers. The sum ji+...+jm of these integers is called the order of D. Thus the unit operator 1 has the order 0 because ji=...=jm=0« Let i,.., xm) (v = 1,2,..., n) be n rational functions with real coefficients, and let Di,..., Dn* be n differential operators such that the order of Dv does not exceed vl (v = 1,2,..., n). The determinant Difi Dif 2 ... D^ / D 2 f a ... D 2 f n \ fi ... Dnfi Dnf2 ... Dnfn is called a generalised Wronski determinant or a Wronskian. This Wronskian evidently vanishes identically when the operators Di,..., Dn are not all distinct. It also vanishes identically if fi,..., fn are linearly dependent over the real field. For an identity cifi +...+ cnfn = 0 implies the n identities
ROTH'S LEMMA .. ... + cnDyfn
79 E
,,..., n).. 0 (v = 1,2,..., If now ci ,..., en are not all zero, then the determinant of this system of linear equations for Ci ,..., cn vanishes, and this determinant is the Wronskian we considering. Let these two trivial cases be excluded. It will then be proved that at leastt one Wronskian of the given functions is not identically zero, at least when fi ,..., fn are polynomials. 4. The case of functions of one variable.
Let
fi, = Mx) (v = 1,2,..., n) be n rational functions in one variable x which have real coefficients and are independent over the real field; thus, in particular,
fn(x) * 0. There is only one Wronskian of these functions that does not vanish trivially, viz. that Wronskian which belongs to the operators d2
d
dn1
We show by induction for n that this Wronskian is in fact distinct from zero. This is obvious for n=l since then
to. Let therefore n ^ 2, and assume that the assertion has already been proved for n1 functions. Put
These n1 functions are still independent. For any equation ClF1+...+CnlFnl

, with real coefficients implies, on integrating, that cifi+...+c n lfnl B where cn is a further real number, whence ci = ...=cnl=Cn=0 because fi ,..., fn are independent by hypothesis. It follows then from the induction hypothesis that
(?;:::£:!)
*0
Next one easily shows that, for any rational function g, identically
)•&*(
80
LECTURES ON DIOPHANTINE APPROXIMATIONS
Here choose g=fn~1. Then in the Wronskian on the lefthand side all but the first element of the nth column vanish, and this determinant reduces to \jug... ins/
\*i« Ani/
* 0.
Hence, finally, /^Di... Dm _ I DI ... DH!\f~tt ± n
V fi ... fny
\Fi... Fniy
'
whence the assertion. 5. The general case.
From now on let
ri rm /\ fp(xi,..., xm) = L ••• Li *ii—im Xll « xm ii=0 im=0
(^ = 1,2,..., n)
be n polynomials in xi,..., xm that have real coefficients and are independent over the real field. We want to show that at lea'st one of the Wronskians in these functions is not identically zero. In the special case m=l this assertion has just been proved, even for the more general class of rational functions. To reduce the general case to this special one, denote by x a new variable, by g a positive integer exceeding all the degrees ri,..., rm, and put YI
ffn
U=0
im=0
1
(v = 1,2,..., n). ?
m
1
The exponents ii+iag+i3g +...+img "" .of x maybe considered as representations to the basis g, with ii, i2,..., im as the digits; for by the choice of g these numbers may assume only the values 0, 1, 2,..., g1. Since there is only one representation of any integer to the basis g, it follows that no two terms in the multiple sum for ^(x) are constant multiples of the same power of x. This implies that xmi)Sj,(xm) v=l where Pi,...,Pn are polynomials in xi,...,xmi and Si ,...,2^ are polynomials in xm, all with rational coefficients. From now on choose one such representation for which the number n of terms is a minimum; then Let us call this the minimum representation of A. In the minimum representation, both the n polynomials Pi (xi ,...,xml),.., Pn(xi ,..., and the n polynomials are independent over the rational and hence also over the real field (§2). For assume, say, that there are rational numbers ci ,..., cn not all zero such that ciPx +...+cnPns°J let eg> cn+0. On solving for Pn, Pn s yiPi+—+ynlPnl where yi,...,ynl are rational numbers. Hence we obtain a new representation of A, n1 ^ # A(xl9...,xm) = 2) Pl/l(x1,...,xmi)S (xm) where S (xm) = 2v(xm) + rvZn(xm. v=l , with at most n1 terms, contrary to the definition of the minimum representation. The independence of 1^ ,...,£n is proved in the same way. By Lemma 1 there exist then two Wronskians
that do not vanish identically. Here, in the Wronskian U*,
ROTH'S LEMMA
83
*
with certain nonnegative integers jpi,..., Jy m l On the other hand, in the Wronskian V**, Ji/m D where * = i km^l
G> = 1,2,..., n).
Denote by D^ and A^j, the new operators * ** 9 ll 3  xml m and
Further put
D12A ... DmA D2iA D22A ... D2nA ... DnnA ••• AmA
A21A A22A ... A2nA An2A ... AnnA Thus W(xi ,...,xm) = C W*(xi ,...,xm)
where C=K) is a certain rational number. On differentiating the minimum representation of A, we obtain the system of identities n Therefore, by the multiplication law for determinants, W*(xi,...,xm) = U*(xi,...,xmi)V**(xm) and hence also = CU*(xi,...,xm_i)V**(xm). It is obvious that all three determinants U*, V**, and W are polynomials with rational coefficients in some or all of the variables xi,..., xm. Moreover, the stronger result holds that W has integral coefficients. For if ji,..., jm are arbitrary nonnegative integers, the partial derivative
84
LECTURES ON DIOPHANTINE APPROXIMATIONS
has the explicit form 11ii ji ... ximJm m
and hence is a polynomial with integral coefficients. On the other hand, the general element in the determinant W is exactly hence is such a polynomial, and so the same is true for W. Now a wellknown theorem due to Gauss states that if f and g are polynomials in any number of variables with rational coefficients such that the product fg has integral coefficients, then there exists a rational number c±0 such that both cf and c"1g have integral coefficients. On applying this theorem to the two polynomials CU* and V**, we find that there are two rational numbers jx+0 and w¥Q such that U(xi,...,xmi) = uU*(xi,...,xmi) and V(xm) = vV**(xm) have integral coefficients, and that further W(xi,...,xm) = U(xi,...,xmi)V(xm). The following result has thus been obtained. Lemma 2: Let ri
r
5f
ii=0 im=0 be a polynomial with integral coefficients. There exist a positive integer n not greater than rm+1 and a system of n2 operators A
]LB/=
where j^i,..., jpiml* Ji/m are nonnegative integers such that and that the following properties hold. The determinant AH, A Ai 2 A ... Am A A 2 iA A 22 A ... A 2n A A
mA ^A
... AnnA
does not vanish identically, is a polynomial with integral coefficients, and can be written as a product = U(xi,...,xmi)V(xm) where U and V are likewise polynomials with integral coefficients.
ROTH'S LEMMA
85
7. Majorants for U,V and W. If 1 1 ^ v? A(xi,...,xm) = E ... E ai ii=0 im=0
and
ri rm B( Xl ,...,x m )= E  E ^ ii=0 im=0
are two polynomials with real coefficients such that 'aii ...im I * bii ...im
for
all suffixes ii ,..., im,
then B is said to be a majorant or majoriser of A, and we write A«B. It is obvious that this relation has the following properties. // A « B and B «C, then A «C. If A « Band C «D, then A*C «B +D and AC «BD. If A < < B, and c is any real number, then cA« c B. The relation A«B may be differentiated arbitrarily often with respect to any of the variables. We also use the notation,
and call [A] the height of A. This agrees with the definition of the height of a polynomial in a single variable given in Chapter 3. We consider now again the polynomial ri rm . A(xi,...,xm) = Z ••• E aii...imXl1— *r& ii=0 im=0 of Lemma 2 and denote its height by a= Al. By the binomial theorem, Hence A has the majorant A(xi ,..., On differentiating this formula repeatedly, we find that
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LECTURES ON DIOPHANTINE APPROXIMATIONS
Here
so that In particular, it follows that AMy APmThis index may also be obtained as follows. By Taylor's formula,
88
LECTURES ON DIOPHANTINE APPROXIMATIONS
ri rm + _ + Pm E  Z Ajl...jm(/c1,...,Km)xpl xijl...xm3m, ji=0 jm=0 where x is a further variable. Hence J(A), for A^O, is the exponent of the lowest power of x in this development with a nonzero factor
=
A
3i —Jm^1 »••• From this, it follows at once that if B(xi,...,xm) is a second polynomial of the same kind, then (A): J(ATB) £ min{j(A), J(B)}, (B): J(AB) = J(A) + J(B), where the indices are taken at (KI ,..., /cm) relative to pi ,...,pm< obvious that
K is
further
either J(A) £ 0 or J(A) = «>, and thai? J(A) = 0 if and only if A(/d ,...,Km) * 0. We need one further simple property of the index. Let li ,..., 1m be arbitrary nonnegative integers, and let Evidently B.. . («!,..., «m) =H1 )...(? m ) A.* .*(Ki,.. 0 Km) Ji0m \Ji/ \3m/ ji—Jm
where m . m .A m Z 7^ = Z J  Z ^h=l ph h=l p h h=l p h Since the index cannot be negative, we obtain then the inequality * ji*=ji+U,...,Jm = im+lm and therefore
(C):
J(Ai1 ...i
m
) ^ max(0, J(A)  Z ph ^ )• h=l
From now on the index J(A) will nearly always be taken relative to
2. Put w(A) a e"J KI,...,KM are Positive integers, the ordered system of numbers b, si,...^]^, KI,...,KM is said to have the property TM V either, (i) M = l,
ROTH'S LEMMA
91
or, (ii), simultaneously M**2 and
Kh > Si logKi
(h = 1,2,..., M);
isit b ^ KiM Therefore the following inequalities also hold, M h=l By way of application, let m^2; let the ordered system of numbers a, ri ,..., rm, Hi ,..., Hm have the property Fm; let n be an integer such that 1 ^ n ^ rm + 1; and let i m b =2 — n!a n ; Pi,= nri, P2= nr2,...,pmi = Then the new ordered system of numbers b, pi,.«»Pml> Hi,...,H m _i has the property r m _i. Proof: The first inequalities
phlog Hh ^pilogH! (h = l,2,...,ml are for m^3 immediate consequences of the assumption that
(h = l,2,...,m). Next, by hypothesis, w ^ o Y(ml)m(2nnl) Hi ^2 ,
whence, trivially, also
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LECTURES ON DIOPHANTINE APPROXIMATIONS
Finally, n « r m + l ^ r i + l « 2ri, hence n! ^ nn ^ 2nri = 2P1, and, by assumption, a«Him
T
f
, ri + r 2 + • • • + r m
Therefore,
D ^ z92nmri ^pi «an ^ & (2m+l)p1 because
1 , 1 .
1
12. A recursive inequality for &m. I. Let again t be a constant such that 0 r i>«» r m>Hi,...,H m ) = J(A;ri,...,rm; Ki,...,/cm). Denote by n the integer with Kn and by U(xi,...,xml), and W(xi ,...,xm) the three polynomials that correspond to A by Lemmas 2 and 3, and put again b =2 ' ' n ! a11;Pl = nn,...,pmi = nr m _i, p m = nrm. As has just been proved, the ordered system of numbers b,pi,...,pm_i, Hi,...,Hmi has then the property rmi. From the construction and from Lemma 3, 1X1 < a; ful < a, IvI ^ a, (wl ^ a, where
ROTH'S LEMMA
93
Therefore the upper bounds for the degrees of U, V, and W imply that U(xi,...,xmi)eR(b;pi,...,pml), V(xm)eR(b;pm), W(xi,.,.,x m )eR(b;pi,...,p m ). Hence, in particular, with the same fractions KI ,..., K m l> Km as above, J(V;pm;«m) * Si to Pm;Hm). From the identity W(xi,...,xm) = U(xi,...,xmi)V(xm) and from the multiplicative property (B) of the index, it follows that J(W;pi,...,pm; i»...> ffm) = J(U;pi l ...,pmi;«i l ,..,jeml) + J(V*;pm;«m), or
where, for shortness, $m = ®mlft>;pi v*Pmi;Hi,...,H m i) H 81 (b;pm;Hm). Instead, we may also write because ph = nrh for all h, and so, by the definition of the index, J(W;ri ,..., rm; KI ,..., fm) = n J(W;pi ,...,pm; «i ,•> «m)Since from now on only indices of polynomials at the fixed point (KI ,...,/tm) relative to the fixed integers ri ,..., rm will occur, we shall write for these indices simply J(W), J(A), etc. 13. A recursive Inequality for
®m. II.
In the inequality
J(W) ^ n$m just proved, we can give a lower bound for J(W) in terms of J(A). For, as in §7, AA.A
A,
where the systems of suffixes /*!,..., jnn run over all n! permutations of 1,2,. ..,n, while the operators Ay are of the form
and the j's are nonnegative integers such that 3Ml+...+j]Liml « Ml,
3,/m = v1
(lJ>,v = 1,2,. ..,n).
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LECTURES ON DIOPHANTINE APPROXIMATIONS
Therefore A^A = Aj/Jll...]1Lim_1jj;m,
so that property (C) of the index implies the inequality J(A^A) > max(o, J(A)  *£ j \ h=l h But and so m 1
\
" h=l
^ maxfO, J(A)  **"  ^Therefore, finally, by the properties (A) and (B) of the index, max (o, J(A)t^ ). r \ m/
This inequality can be simplified. For shortness put N = [{ J(A)t}rm] + 1, where [x] denotes as usual the integral part of x. Hence Nl«{j(A)t}r m N. The case n^ N. Evidently nKN1 Hi ,..., Hm1 with the property rmi such that 0m(a; ri ,..., rm; Hx ,..., Hm) « t where
14. Proof of Roth's Lemma.
It is now easy to prove Roth's Lemma: Put cm=2m 3. If the ordered system of numbers has the property rm, then Cmt Proof: We procede by induction for m. First let m=l, hence Hi ^ 2 and a^HF1*. The estimate ( 0i) of §10 implies then that
as asserted. Secondly assume that m^2, and that the assertion has already been proved for all ordered systems of numbers b, Pi v>Pml» HI ,..., H m _i with the property rmi; it suffices to prove that it then is true also for all ordered systems of numbers a,n,...,rm,Hi,...,Hm with the property rm. By this induction hypothesis, the expression ^m in Lemma 4 satisfies the inequality l(b;Pi,—,Pmi;H1,...,Hmi) +t < cmit2 "(m"2)
ROTH'S LEMMA
97
and therefore Lemma 4 implies that 0m(a;ri,...,rm;H1,...,Hm) ^ t Now 0 + l2(">2> )
whence the assertion. We conclude this chapter by stating Roth's Lemma in a slightly weaker, but more convenient explicit form, as follows. Theorem 1: Let 0 1, Fo * 0, Ff * 0 and therefore F(0)*0. We impose the additional condition that F(x) has no multiple factor, hence that F(x) and its derivative F1 (x) are relatively prime. Let & be an arbitrary (abstract) extension field of the rational field F in which F(x) splits into a product of linear factors The f zeros «=Si,..., If of F(x) are thus all distinct and different from zero. We use the abbreviation c = 2max(F0,F1,..., Ff) so that c ^ 2 is an integer. Lemma 1: For every exponent 1=0, 1, 2,... there exist unique integers gft gi(1),».,gf!:i such that 98
THE APPROXIMATION POLYNOMIAL
99
Fj «J, = gP + gPl ^ + ... Hgj^ { J;1 ty, . 1,2,..., f),
(1): (2):
Proof: First, the coefficients g are unique because the Vandermonde determinant
does not vanish. Secondly, the equations (1) hold trivially for 1 < f1 with g; =F0 and the other coefficients equal to zero. Third, for
and therefore
so that the coefficients are integers. Finally, c1 if 1 « f1,
gPl ..... Ig^l) if 1 >M whence the inequalities (2). 3. A lemma by Schneider.
The following lemma is essentially due to Th. Schneider1 . The proof is taken from Cassels* book on Diophantine Approximation. In the appendix, an entirely different proof is used to prove a stronger result. Lemma 2: Let n ,...,rm be positive integers, and let & be a positive number. Each of the two systems of inequalities m . 0 ^ ii ^ri,...,0 ^ i m ^ r m , E rh £ *Z«(mB) h=l 1. J. reine angew. Math. 175 (1936), 182192.
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LECTURES ON DIOPHANTINE APPROXIMATIONS and
in ^ 1 0 < ii ^ ri,...,0 ^ im < rm, V L — h=l r h
has at most s solutions in sets of integers (ii ,..., im). Proof: The two systems of inequalities are changed into one another by the transformation
(ii ,..., im)—> (ri~ii >•••> r m~^m) and so have the same number of solutions. It suffices therefore to consider the first system. The proof is by induction for m. First let m=l. The system has no integral solution if s>l, and it has not more than
such solutions if s < 1; hence the assertion holds in this case. Secondly let m^2, and assume the lemma has already been proved for inequalities in m1 unknowns. We may assume that because the assertion is trivial otherwise. For fixed i=im> where OM 2j ^7" ^ and so, by the induction hypothesis, has not more than
possibilities. Hence, on putting
er
m
TI .2 r
the original system has at most
m
THE APPROXIMATION POLYNOMIAL integral solutions (ii,..., i m l> *)• be shown that
The
101
assertion is therefore proved if it can
cr^ 1. Now
rm
v
s
i^osi+L
_i r v m r . 2
i^o
Bi+L
. i
T
ft/__
V sa < v S = Z/ L — oT"o ^ 2 2.(i_fL.)2 i=0 is=0 i=0 ss
j\
V ..
s4
s
x ^IT
whence, by s s2 ^ /m1
2m
/ 4m 2 4m
4. The construction of A(xi,...,xm). I.
As before, let ri,...,r m be positive integers. Let further a and s be two positive numbers such that (3): a^l, where f is the degree of F(x). A polynomial of the form
s^4fV2m,
ri rm . x B( Xl ,...,x m )= S  E bii...im i "^m ii=0 im=0
is said to be admissible if, (i) its coefficients b^ f _i m may assume only the [a] + l values 0,1,2,..., [a], and further, (ii) b
ii ...im = °
unless
\ (ms)< E ~^ < \ (mw).
From Lemma 2, it follows immediately that the condition (ii) demands the vanishing of not more than
of the (ri + l)...(rm + l) coefficients of B. Hence not less than
of the remaining coefficients of B may still run independently over [a] + 1
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LECTURES ON DIOPHANTINE APPROXIMATIONS
distinct values. It follows then that there are not less than
admissible polynomials. 5. The construction of A(XI ,...,xm). II. As in the last chapter, put ii
Jl J
1
v
 m
9il+"'+lmB(xi....,xm)
* '"•'
m/ 
* IT
i™
Ji!...Jmia*3l...ad»
Then Jl
'"Jm
'
i1=0 im=0
lm m
" \*lJ
\W
has nonnegative integral coefficients if B is admissible. The same estimate as in §7 of last chapter leads to the majorant B
Ji ...Jm(xi ''
Xm)
« a'2ri *"'+rttlO+»i )ri ...
and hence to Here so that B
Ji Jm^Thus, for all nonnegative suffixes ]i,..., jm,
say, 1=0 is a polynomial in one variable x with nonnegative integral coefficients fc not greater than .
and of degree not exceeding By Lemma 1, it follows now that
THE APPROXIMATION POLYNOMIAL r1+...+rm , PiK»+r m t1 B ( ° Ji...lm V.....V " # #
F
103
(]) ri+...+rml (1L* _ * ** "
L
where
*
1=0
Hence
1=0 «
/\k
k=0 * because FoM«c. Therefore, for all suffixes ji,...,Jm and 0,
T
Here B^
is an integer since j3J^ and g^ are integers. Each number B^
has then at most 2[2a(4c)ri++rm] + i * 5a(4c)ri+"'+rm possible values, and the set of all f coefficients
of
{5a(4c)ri++rm}f possibilities. Let (ji,..., jm) run over all systems of integers satisfying m
jh < 1
by Lemma 2, there are not more than
s
"""
4f
such systems. The corresponding set of integral coefficients
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LECTURES ON DIOPHANTINE APPROXIMATIONS
B*., where 0 = 0,l,...,fl; 0 < Ji< PI,...,O < ] m
M*. There are then more admissible polynomials B than corresponding sets of coefficients B 1 . Hence there exist two distinct admissible polynomials 9 _ ri rm B(Xl,...,xm)= 2 ... ii=0 and ^B(xi,...,xm)= £ ... £ bi^.,1mxl1...xjg1 ii=0 im=0 with the following property: Define integers JW and B'
such that
"' v £ ^
t} ^
=
I 0=0
; f ^ * \ (ms) ^ n=1 rh
THE APPROXIMATION POLYNOMIAL
105
Put
ri rm Z ••• E ii =0 im=0
Since B and B are distinct, From the construction, the coefficients a^ ^ values not exceeding a, thus satisfying
of A are integers of absolute
a ij _ im N5(4c) ri+  +rm . Moreover, 1
...lm
1 2
£?ih . ~. rh n=l
1, 2
'
and furthermore,
m *. * ,^)0 if ^^l,2,..., Instead, we may also say that AJ^.J^XJ...^) is divisible by F(x) whenever m j, 0^ j i ^ r i , . . . , 0 < ] m ^ r m , 2 ^^ 5 (ms); h=l r h * for the zeros £1,..., ^f of F(x) are all distinct. We note that the upper bound for the coefficients of A implies again majorants analogous to those found for B. The following result has thus been proved. Theorem 2: Let F(x) = Foxf + Fix1"1 +...+ Ff, where f * 1, F0 =1= 0, Ff * 0, be a polynomial with integral coefficients which has no multiple factors and does not vanish for x=0. Put c = 2max(F 0 ,Fi,..., Ff). Let ri,..., rm be positive integers, and let B be a real number not less than 4fV2m. There exists a polynomial ri rm . A( Xl ,...,x m )= E .. Z aii...imxi1...xmn*0 ii =0 im=0 with the following properties. (1): Its coefficients ailB..im are integers satisfying
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LECTURES ON DIOPHANTINE APPROXIMATIONS
and they vanish unless
n=i (2): Aj 1>a< j m (x,...,x) is divisible by F(x) whenever m j. . 0 < h < r!,...,0 ^ jm ^ rm, Z rh ^ ^ Z«(ms) h=l (3): The following majorants hold,
A
Ji...Jm(x''x> This theorem will be applied only for large values of m, and s will always be small compared with m. The last two majorants hold, of course, by the formula proved In Chapter 5, §7, since in the present case,
Chapter 7 THE FIRST APPROXIMATION THEOREM 1. The properties Ad, B, and C.
While the last two chapters depended on purely algebraic ideas, we now introduce real and gadic algebraic numbers and study their rational approximations with respect to the corresponding absolute value or gadic value, respectively. Here, as usual, g = pf 1 ...pr r *2, where pi ,...,pr are distinct primes, and ei ,..., er are positive integers; the gadic value \A\g of A**+~(ai,...,ar) is defined by
lo The later occurring gfadic and g fl adic values agf and ag" are defined analogously. The letter £ always denotes a fixed real algebraic number, and the letter E a fixed gadic algebraic number. Only £ satisfying and only a"— (£i,...,£ r ) satisfying «i+0,, « r + 0 will be considered. We denote by F(x) = Foxf + Fix*'1 + ... + Ff, where f > 1, F0 + 0, Ff + 0 , a polynomial of lowest degree with integral coefficients having either £, or JET, or both £ and S, as zeros; hence, by Chapter 3, F(x) has no multiple factors. As before, we put c = 2max(F 0 , Fi,..., Ff), so that c > 1. Next we denote by
s= {jcW JC W JC W,...} a fixed infinite sequence of distinct rational numbers
(k)
(k)
" where P
% 0 and Q
* 0 are integers such that (P(k),Q(k)) = l.
107
108
LECTURES ON DIOPHANTINE APPROXIMATIONS
We call H = max(P< k >UQ (k >) the height of *(k). It is obvious that lim H(k) = oo. k*» For such sequences E we now define three properties A^, B, and C where d is either 1 or 2 or 3. First, 2 is said to have the property Ad if for d=l: There exist two positive constants p and ci such that
(1):
U ( k ) Sl^
(Ai):
ClH
(k) p
'
for all k;
for d=2: There exist two positive constants or and c2 such that U(k)£lg « c2H(k)~a
(A2):
for all k; and
for d=3: There exist four positive constants p, a, ci, and c2 such that (A,): k (k) * Cl H (k)  p and U W ff g « erf00"* for aU k. The property As includes therefore both properties AI and A2 . If S has the property Ad, then its elements have for d=l and d=3 the real limit £ > and for d=2 and d=3 the gadic limit or, because CiHfc)p and c2IT ^"^tend to zero as k tends to infinity. Secondly, £ is said to have the property B if there exist, (i) two integers gf and gft satisfying g' £2, g" £2,
(g',g")=l;
(ii) two real numbers X and ju satisfying
0 < A < 1,
0 < p < 1;
and
(iii) two positive constants c3 and c4, such that (B): P W  g( *c,H (k)X  1 and Q(k)g» * (^H^'1 for all k. The first inequality (B) holds trivially if A=l as we may simply take c3 = l; and similarly for the second inequality when ju=l. For later it is important to note that if d=2 or d=3, and if S has both properties Ad and B, then
(g, g1) = 1 if 0 ^ A < 1. For lim P(k)L, =0, while (P(k),Q(k)) = 1, hence Q(k)Li = 1, and so also k—»°° ^ ° lim l/c(k)L, =0.
k— »oo
f
&
If now g and g had a common prime factor, pi say, then
THE FIRST APPROXIMATION THEOREM lim kl =0 k>«> Pi
109
a n d l i m U  { i L 1 = 0 , hence £ i = 0 , k*°o P
contrary to the hypothesis. Third, S is said to have the property C if there exists a positive constant c5 such that (C): U^N c5 for all k. In the two cases d1 and d=3 the property C follows from the property Ad because
In the remaining case d=2 it is, however, independent of AdOur first aim in this chapter is to prove the following result. Main Lemma: If the sequence has all three properties Ad, B, and C, then r ^ A + p, where rp if d=l, (2): T ~ J or if d=2, LPKT if d=3. The proof of this lemma will be long and involved, and it will be indirect. It will be assumed that (3): r = A.+/n4e where e > 0, and from this hypothesis we shall deduce a contradiction. 2. The selection of the parameters. Since the property Ad weakens when the exponents p and a are decreased, we may without loss of generality assume that (4):
0 < e ^.
For the same reason we are allowed to assume that (5): ci & 1, c2 > 1, ca ^ 1, c4 ^ 1, c5 ^ 1. Similar to c, CI,...,CB the letters c fl , c 7 ,...,Ci, C2, C3, TI, T2, and T8 will be used to denote certain positive constants that depend only on the sequence S and the algebraic numbers {, S, or £ and S9 respectively; they will, however, be independent of the numbers m, s, t, Ki,...,K m , ri,..., rm
110
LECTURES ON DIOPHANTINE APPROXIMATIONS
to be defined immediately. The last three constants TI, T2, and T3 will not be fixed until the end of the proof. The parameters are now selected as follows. First, choose a positive integer m such that
and in terms of m define the positive number s by (6):
s=f.
Secondly, choose a number t such that
(7):
1 otoD 0 < t * 1, 2m+1 12 « SL .
Third, select m distinct elements K^ll\ K^,...,K(im) of Z that satisfy certain inequality conditions to be stated at once. To simplify the notation, these elements of S are written as
where the Ph and Qh are Integers for which Ph * 0, Qh + 0, Thus Kb has the height Hh = max(Phl,Q h l). The hypothesis of the main lemma imposes, for all suffixes h=l, 2,..., m, the following inequalities:
(Ad):
~Uh$l'sc1Hnp
ifd=l,
< UhSlg^caH^
if d=2,
J Kh~ ? I *CiHhP and I «hslg«ca Hnff (B):
if d=3;
1
Phlg' scaHh" and
(C): It is necessary for the proof to add the following conditions:
(8):
Phg' « g V i f O < X < l
(9):
log Hh+1 *  log Hh
and, depending on the suffix d,
(h= l,2,...,m), (h = 1,2.....m1),
THE FIRST APPROXIMATION THEOREM
111
/ £m2 £(ml)m(2m+l) \ Hi £ maxV(20c)t , 2l , Td/ . Since the elements of S satisfy the limit formula (1), it is possible to choose Ki,...,/cm such that all these inequalities are satisfied. Finally, select m positive integers ri,..., rm such that O Irw* TI ri & 21ogH m ri e log Hi ' (10):
(12):
rh ^ ri Jjjj > ^
(h = 2,3,..., m) .
Since, by (9) and (10), evidently 2 < Hi < H2 < ... < Hm, these formulae imply that
because, by (4),
Hence we find that (13): rilogHi ^ rhlogH h ^ (l+e)rilog HI (h = 1,2,..., m). Therefore, for arbitrary non negative exponents ki,..., km, it follows that
We also note that, by (9), (11), and (12), rh *  > 2, rh1 > 2rb
2rh+1 logHh+l < rilogHi ^ rh l°S Hh,
hence
rh+1 < 2 log Hh < t rh logHh+1 " ' and therefore (15): Thus, trivially, (16): ri > ra >... >rm
r h + i< rht and
(h = 1,2,..., m1).
ri + r2 + ... + rm
1 depending only on S such that IK
Ig < c8
for all k,
and therefore also l5lg« c8. This time we substitute the values in the identity (27), so obtaining the equation ri rm /\ A \ i i (31): A(1) = £ ... £ A1 1 (S,...&fa}J m)(/c15)]l"ll...(/cm5')3m"lm. ji=0 jm=0 Jl"°m w ^lm/ Here, just as in (28), it suffices to extend the summation only over all systems of suffices (j) = (ji,..., jm) in J. The binomial coefficients in (31) are integers, hence their gadic values are not greater than 1. It follows then from the nonArchimedean property of the gadic pseudovaluation that (32): A (1)  g oo
and hence there is a positive constant GO such that ko Ig ^ c»
for
^ k«
From (46) and the identity
it follows finally that , where c = c0 I*'1 g ca. We apply now the main lemma to the sequence So instead of S and find that a ^ jit + X, giving the assertion. g
10. Polynomials in a field with a valuation. The second form of the First Approximation Theorem makes a statement on the values of a polynomial assumed in a sequence of rational numbers. Before enunciating and proving this theorem, it is necessary to discuss first a property of fields with a valuation. Let K be a field with a valuation w(a), and let Kw again be the completion of K with respect to w. We say that K has the property D if the following compactness condition is satisfied: (D): Every infinite sequence of elements of K that is bounded with respect to w contains an infinite subsequence which is a fundamental sequence with respect to w, hence has a limit in Kw. Let K have this property D, and let F(x) = F0xf+Fixf"1+...+ Ff, where f > 1, F0+ 0, be a polynomial with coefficients in K which has no multiple zero in Kw. Put G(x) = Fo1F(x) = xf+GiXf1+G2xf2+...+Gf, y= t+w(Gi)+w(G2)+...+w(Gf), so that
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LECTURES ON DIOPHANTINE APPROXIMATIONS
and also G(x) has no multiple zero in Kw. Assume now that x is an element of Kw such that w(x) > y
and hence
w(x) > 1.
Then w(G1xf1+Gaxf2+...+Gf) ^ (yl)max(w(x)f1, w(x)f2,...,w(x), 1) < < (y Dwtx)*'1, and therefore w(G(x)s>w(x*)  w(Gixf1+G2xf"2+...+Gf) ^
Conversely, it follows that if
w(F(x)) < w(F0), then w(x) «y,
because the first inequality implies that w(G(x)) = vrCFo1 F(x)) ^ wfFj'MFo) = 1. Consider now an infinite sequence 2 = {ic'1', ic2', K'S',...} of elements of K satisfying lim w(F(/c(k))) = 0. This assumption implies that )
)) < w(F0)
and hence
for all sufficiently large k. Thus the sequence 2) is bounded with respect to w and so, by the property D of K, it contains an infinite subsequence S1 ={/c(il), /c(ia), /c(is),...}, where i1°° We now apply Lemma 21 repeatedly, once for each prime factor p] of g. First, there exist an infinite subsequence Si = {K , K , JT S, apiadiczero £1 of F(x), and a constant y(0 >0, such that
,...} of
for all k. 2i) Secondly, there exist an infinite subsequence S2 = of Si, a p2adic zero £2 of F(x), and a constant y W > 0, such that
p2
forallk,
I ) pj
for aU k.
...
while, naturally, also
Continuing in this manner, we obtain for every suffix j=l, 2, ...r an infinite sequence
Sj
= frM, K(^\ fc(hi3),.}, where
Si 2 S22 ... 2 Sri apjadiczero gj of F(x), and a constant yj > 0, such that K (hjk) *ilpi^ y ft) lF&c (h l k) )lii l Let S
tff
for i=l,2,...,j andfor all k.
be the sequence Sr; further put / logg (1)eilogpl maxU maxyy
logg \ (r)e r logp r / ) , ,..., y
and denote by E the gadic number S —(?!,...,
r),
which is algebraic and a zero of F(x). We have then
max
logg \ ^^^lg10^1^
f logg 1 (i) h 1OgPi max ^ (y F(^ rk))lp> f i=l,2,...,r ^ j for all k
and hence
for all k, whence the assertion.
130
LECTURES ON DIOPHANTINE APPROXIMATIONS
12. The property A'd. As earlier in this chapter, let again F(x) = Fox* + Fix*'1 + ... + Ff, where f > 1, F0* 0, Ff * 0, be a polynomial with integral coefficients which does not vanish at x=0 and has no multiple factors, hence also no multiple zeros in any extension field of the rational field. Further denote again by $ a real zero and by S a gadic zero of F(x), and by p and or two positive constants. Finally let again S = {ft'1', /c'2', /r3',...} be a sequence of distinct rational numbers «W  5jjJ + 0
of heights
H = maxdP^I, Q(k>)
such that P ( k >*0,Q ( k ) +0, (P 0 , a>0, 0 < X < 1, O ^ j u t ^ l ; ! let ci, c 2, c8, c4 be positive constants; and let gf & 2 and gM ^ 2 be fixed integers. Finally let S = fro1/ K\2\ K^,...} be an infinite sequence of distinct rational numbers K(k) = ?
* 0, where P(k)+ 0, Q(k)+ 0, (P(k),Q(k)) = 1,
with the following two properties. (Aj): For all k, t/d=l, ) ff

t/d=2,
132
LECTURES ON DIOPHANTINE APPROXIMATIONS F(ie(k))l * ciH(k)p and F(*(k))g * ciH(k)a
if d=3.
(B): For all k,
and Then for d=l,
v ^ \+p,
for d=2,
p+a ^ A. + JU
for d=3.
Proof: It suffices to apply the first form of the theorem to the sequence 2' and the zero or zeros $ , E obtained by Lemma 4. By the same lemma, the new second form of the theorem implies also the original first form; both forms are thus equivalent.
Chapter 8 THE SECOND APPROXIMATION THEOREM 1. The two forms of the theorem.
This chapter contains a generalisation of the First Approximation Theorem which has just been proved. We begin by introducing some notations that will be used. If a is any real number, and /3 is any padic number, put I a I* = min(a, 1), so that always
/3* = min(j8p, 1),
Denote by P1,p2,...,pr; Pr+1»Pr+2»">Pr+r»;pr+r'+l'pr+rf+2'"'' pr+r'+r" a fixed system of r+r'+r", =n say, distinct primes. It is not excluded that one, two, or all three of the numbers r, rf , and rlf , are equal to zero. Let further £*0, £i+0,..., £ r +0
denote a real algebraic number, a piadic algebraic number, etc., a pradic algebraic number, respectively. These algebraic numbers need not satisfy the same irreducible algebraic equation with rational coefficients, and thus they may belong to different finite extensions of the rational field. Next let F(x), Fi(x),..., Fr(x)
be r+1 polynomials with rational coefficients, which neither vanish at x=0 nor have multiple factors. It is not required that all these polynomials are distinct, that they are irreducible, or that they are nonconstant. As in previous chapters, let again S ={/cv/, tf(2), «s3),...} be an infinite sequence of distinct rational numbers ,« = Z^
+0,
where P(k) +0, Q(k) *0, (P(k),Q(k))  1, H(k)=max(P(k),Q(k)).
Finally, put pj
133
PJ
134
LECTURES ON DIOPHANTINE APPROXIMATIONS
and
r j=l
r+r* *
J
/ \
j=r+l
r+r'+r" 3
]=r+r f +l
3
and denote by Id, Ka, and r three positive constants. The Second Approximation Theorem can now be stated in two different, but equivalent forms, as follows. Second Approximation Theorem (I): If for all /r e s,
then T« 2. Second Approximation Theorem (n): If for all /r 'cS, * ri>..,rr such that, for all k,
and
(2):
(FjOc(k))  J < yj U(k) gj *j
(] = 1,2,...,r).
Consider, for instance, the inequality (1). If E contains no infinite subsequence S such that (3):
lim ic^k)g* = 0,
then the greatest lower bound M=
THE SECOND APPROXIMATION THEOREM
135
is positive, and it is obvious that for all k, F(« ( k ) )l**rl« ( k ) «l*
where
y£»l.
Next assume that S does contain an infinite subsequence £' with the property (3). Then £ is a zero of F(x), and hence where G(x) denotes a certain monic polynomial with real coefficients. Denote by yo > 1 a number such that G(x)  < y0 for all real x such that xg I < 1 and hence x < U 1+ 1. We have then, for every k. either
U^4 I > 1
or
1
and hence, trivially,
F(/c^) %y 0 k^{
and hence
F(«(k))

r+r f +r M /, v j=r+r ? +l
and hence
j=r+l Next put
Then, for all k,
and
because
s2
is an increasing function of s, and 0<e^l. It is, in particular, (k) evident from these relations that all a j are bounded, s
Q ^ a^
*$ a;
here a is a certain positive constant that does not depend on k.
142
LECTURES ON DIOPHANTINE APPROXIMATIONS
Let N be the positive integer
so that 1 + £ log Pj < V 3=1
Further put
Since 0 « e(jp < [a N],
the system of the n+1 nonnegative integers \6o
9
ei ,..., en /
has not more than possibilities. Hence there exist an infinite subsequence S' = {/c(il\ K(l2*; K^l3\...} of S, where u < i2 < is e lo
N
0
j=l
f , ^*N
e
o + L e j lo gP] > "H" 3=1
just proved are equivalent to (7): (l ^)N < eo + logfeg' g f f ) « N,
e0 + logg > ^
> 0.
They imply that at least one of the two numbers p = —j^2,
(8):
o =
1^
,
which evidently are nonnegative, must be positive. By §5, we have fl
IK
\
)e
—c 
/I \
•"• ii
(lf\
*•*/
/I \
=
(]r\ (lr\
/« V /OV
JjL
/
/lr^
R' 'p«
^ id
**
IN
^lr\
^ Jti
(24eW
V*''^C/C0
N
This result may be strengthened to (9):
U (k) £l ^H (k) " p
ifp>0.
/ '
=
H
144
LECTURES ON DIOPHANTINE APPROXIMATIONS
For, by hypothesis, H (k) £ 2 and hence H(k)"p < 1. Similarly, again by §5, for ] = 1,2,...,r, _..W. (k H >
N
(2K)ejlogpj " «„«
Assume, for the moment, that cr> 0, hence that ei,...,er do not all vanish. We may, simultaneously, renumber the primes pi,...,Pr and their exponents ei ,..., er. It may thus be assumed that, say, ej > 0 f or j = 1,2,..., u9 but 6j = 0 f or j = u+1, u+2,...,r. Here 1 < u < r , and g becomes now the product Denote by E the gadic algebraic number S— (Si,...,5u) with only u components; by the hypothesis, none of these components is zero. Just as in the proof of (9) we find that (2+€)ejlogpj ( (k k %lpj*H >~ N (J = 1,2,...,U). Since, by definition, logg
logg
it follows then that (2K)lQgg W
CIO):
U ffg«H
W
N
=H
f f
Next put (II).
VIA;.
x1 lOgg* A  i  (2+6)j^.
,
lOgg" / X  i  (2+6)jq.
It was shown in §5 that r+r1 /. x r+r'+r11 /. x E a^logp^l, E j=r+l J j=r+r'+l Therefore r+r' r+r' /.» /.» ' M r+r' /./. v logg'= E eJr log Pj *N E « Wpj=r E a3 J J J W j=r+l j=r+l s j=r+l
if0.
THE SECOND APPROXIMATION THEOREM
145
and similarly
so that (12):
O ^ X ^ l , O^jii^l.
In particular, the equation X = 1 holds exactly when gf = 1, and the equation /i= 1 when g"= 1. For j = r+1, r+2,...,r+r!, M M to (2+e)ejlogp1 p(k)p. = H(k)ajk)logPj . H(k)«jkVk)logPj * H(k)' £—X
m
Hence, from / logg' logg' \ p(k)g =max^P(k)p^l^Pr+l ,..., p*>P r '> and Q Finally put
ejk) ^ 0 if j = r' + l,r'+2,...,r'+r".
(k)
cannot have prime factors distinct from
cannot have prime factors distinct from Pr' + i »...,p r T +r
154
LECTURES ON DIOPHANTINE APPROXIMATIONS
It Is clear that H*' tends to infinity with k, and that *
p(k)
eeEfe>
0
Pr+r'» Pr+r1 +!>•••» Pr+r'+r1 be finitely many distinct primes. Denote by S an infinite sequence of distinct triplets {p(k),Q(kU(k)}
(k = 1,2,3,...)
where P^ , Q*', and IT ' are integers as follows, 1. See Hardy and Wright, Theory of Numbers (Oxford 1954, 3rd ed.)« 335337; K. Mahler, Mathematika 4 (1957), 122124.
156
LECTURES ON DIOPHANTINE APPROXIMATIONS P» + 0, Q(k) + 0, R(k) + 0, P« + Q« + R(k) = 0, )> Q(k)} = (p« R(k)} = (Q(k)> R(k)}
Put
and write P(k*, Q^\ and R^ as products of integers, P« = P^fP,
Q(k)  Q^OJk),
R(k) =
R^,
where Pi has no prime factors distinct from pT+i,...,pr+ri, Qi no prime factors distinct from Pr+r'+l'—'Pr+r^r1*' a^ Ri has no prime factors distinct from pi , ..., pr . # (k= 1,2,3,...,), z/ ^ 1. Proof: For each k, either
(7):
P (k) IHQ (k) * BWI,
or one of the five inequalities obtained from (7) by permuting P , Q , R is satisfied. Since we may replace 2 by any infinite subsequence, and since we are allowed to rename these three letters and, at the same time, the corresponding sets of primes, there is no loss of generality in assuming that, in fact, the inequality (7) holds for all elements of S. Put now (k) P(k) w and $ = 5i =  = 5r = 1. and, just as in the last chapter, write
n j=r+l
j=r+r'+l
Then, by (7),
,<w
(k)
Further also
,(k) QM
*  RIfi^l Pj 'Pj
For either PJ is a divisor of Q(k*, and then it is prime to R^ and so
APPLICATIONS
157
(IT)
or PJ is prime to Q , and then
R(k)
_ IB R (k)i* Ipj_ iWi IB p] .
Therefore
n U W {jlw n lR (k) L., i—l
*
1—1
and it follows that r
n
/. v r+r? /.v r+r'+r" /. x lipw I n P T>W TT lr*Wp.. IR Ip4. n p.. n Q
3 =1 j=r+l Q Next, the hypothesis implies that, e.g. r
,
(]r\ , D^K'
n IK
r
,
(lr\ , K
,
3
r
^lr^
j=r+r'+l
Tlr^ K
*D> ' 13W ^ Pr+r f +l>— 9 Pr+rf+rM be finitely many distinct primes, and let c and co be two positive constants. Finally let 2 be an infinite sequence of distinct pairs of integers {P(k), Q (k)jguch thatfforall k, P(k) * 0, Q(k> * 0, (P(k),Q(k)) = 1, H(k) = max (  P(k) , Q(k) ) and
j. n IP (k) liPr3
n
lo (k) lD. 0, c3 > 0, a > 0, 0 ^ X ^ 1. If there exists an infinite sequence S = {pW, p(2), P^,...} of distinct positive integers such that, for all k,
then a < X. Theorem (4,n): Let F(x) be a polynomial with integral coefficients which does not vanish at x=0 and has no multiple factors. Let g s* 2 and g1 ^ 2 be two positive integers, and let c2 , c3 , a, and X be four real constants
APPLICATIONS
159
such that ci > 0, c8 > 0, 0, 0 ^ X < 1.
If Mere exists an infinite sequence S = {pW, P< 2 ), P^,...} of distinct positive integers such that, for all k,
then ff < X. Theorem (5,1): Let pi,..., pr, Pr+i,...,Pr+rf be finitely many distinct primes, and let gi * 0,..., £r * 0 be an algebraic piadic integer, etc., an algebraic pradic integer, respectively. Let Ki and r be two positive constants. If there exists an infinite sequence S = {P^\P(Z\P^3\...} of distinct positive integers such that, for all k,
3=1
j=r+l
then r * 1. Theorem (5,n): Let pi,...,pr, pr+l>...,Pr+rf be finitely many distinct primes, and let Fi (x),..., Fr(x) be r polynomials with integral coefficients which do not vanish at x=0 and have no multiple factors. Let KB and r be two positive constants. If there exists an infinite sequence S = {pW, pW, p(3),..} of distinct positive integers such that, for all k, j=r+l then T ^ 1. A discussion similar to that in Chapters 7 and 8 allows again to show that these four theorems are equivalent, in the sense that each implies the other three. It suffices then to prove Theorem (4,1). This is done by essentially repeating those constructions and estimates of §§28 of Chapter 7 that led to the case d=2 of the Main Lemma. One assumes that the assertion is false and that, say, a = X + 4e where 0 < e ^ = . The proof then precedes with the values
and hence with the values = TT » Qh = 1, Hh = Ph
(h = 1,2,..., m).
Here the parameters m, s, t, ri,..., rm are selected just as in §2 of Chapter
160
LECTURES ON DIOPHANTINE APPROXIMATIONS
7, and the polynomial A(xi,...,xm) is defined as in §3. In particular, the inequality (14) of §2 takes the form,
The proof now depends on upper and lower estimates for the integer A
(l)=l fl)=A lx...l m ( ' Cl ''' Cm) * 0 ' which has the explicit value
a
0 .
2. Let again ri,...,r m be m positive integers; let further s, pi,...,pm be m+1 positive numbers. We denote by N the number of sets of m integers (ii,...,im) satisfying the inequalities
(2):
0 * u« n,...p 0 * im * rm, £ j£ *(fs) E JJ , h=1 p h
\^ / h = 1 Ph
or, what is. the same, the number of such sets satisfying
(3):
0 « U « n ..... 0 * im « rm,
That both systems (2) and (3) have the same number of integral solutions is obvious because the transformation (ii >•••> im) * (riii ,..., rmim) interchanges their solutions. 163
164
LECTURES ON DIOPHANTINE APPROXIMATIONS
3. Denote by u a positive variable, and put
and
.ir f=0 m
Evidently, m F(u) = n Fh(u) .
In the sum for Fj1(u) replace i by r^i and note that
rhi _ " " rh =_ /!_" "_ rh \ Ph It follows that
1)
max cosh/u( —  ^2 )) i=0,l,...,rh \\Ph Ph//
.
Now cosh x is decreasing for x ^ Q and increasing for x ^ 0. The maximum is thus attained both when i=0 and when i=rh, and hence Fh(u) « (rh+1) cosh f*£ . Therefore, by (1), m h=l
< (ri+l)...(rm+l) exp that is, (4):
F(u) ^ (n+l)...(rm+l) exp jj ^ f^\1 L h=l V^* /
+^g E h=l
ANOTHER PROOF OF A LEMMA BY SCHNEIDER
165
4. By definition, the inequalities (3) have N integral solutions (U,..., im)« These inequalities may also be written as
and they therefore imply that
On the other hand, all terms in the multiple sum for F(u) are positive. It follows then from (5) that
F(u) £ N exp [ h=l On combining this inequality with (4), we find that
To simplify this estimate, put  m 1 m and fix u in terms of s by ca
The inequality (6) then takes the form (7):
N < (ri+l)...(rm+l) exp jm (M B»   ^t S4 For the applications it suffices to consider values of s with 0 < s ^ n
and hence
s4 < 7 s2 .
It follows in this case from (7) that (8): N < (n+l)...(rm+l) exp (Cms2) where C denotes the expression C  2c* c* °* ~ c2 " 9c ' 5. We finally impose on rh and ph the additional conditions
166
LECTURES ON DIOPHANTINE APPROXIMATIONS
These inequalities evidently imply that
4a GET—©'• a)
4
and hence that
/_9\»
my
2 . 81 UP/ < c? . UP/ 121 ., 3 3^ 121 "TllN5 " ^ T ^ T W = ~8T ^ 2' UO/ UP/
°l«m//121\°
9
It follows therefore that
Thus the following result has been proved. Theorem 1: Let ri,...,rm be m positive integers, and let s, 6e m+1 positive numbers such that
are at most (rnl)...(rm+l)e"ms2 integral solutions (ii,...,im) of the inequalities
0
* ii * ri.....°* »» * r ,55 * S B) h
or, ^o^ is the same, of the inequalities 0 * U « *..... O . l m * rmi ^ JJ Let us compare this estimate with that given by the Lemma 2 of Chapter 6 in the special case when Pi = ri,...,pm = rm ! The notation is slightly distinct at the two places. If we return to that of Lemma 2, then, by this lemma, the inequalities 0 < ii < ri,..., 0 < im < rm, t rh ^ « 2(ms) (or ^2^ h=l have not more than
ANOTHER PROOF OP A LEMMA BY SCHNEIDER
167
integral solutions, and by Theorem 1 not more than
It is easily verified that always
r*\2
^e Wm / 1 such that, for all k,
Hence if X is any sufficiently large positive number, there is an element K ' of 2 for which H^H^^H04. From now on we put for shortness and denote by m a very large positive integer. We further put w 9 ml m 2
t = e" 
,
 m8 X = et
and note that e (H) is given by e(H) = 5a(logloglogHr F . 2. By hypothesis S contains infinitely many distinct elements K , so that lim H^ = oo . k*oo
A THEOREM BY M. CUGIANI
171
It is therefore possible to select m elements
(h=1 2
' .....m)
of S, of heights Hh = H(ih) = max(Phl,lQhl)> ee, such that Xh^Hh^Xjj4
(h=l,2,...,m),
where
2
20^
2
20^
2
2c^
Xi  X, X2= Hif ^ XX * , X8= H/ ^ X2t ,..., Xm = Hm.i
It foUows that
whence, in particular, Hr< H2< ... ra> ... >r m , £ rh* h=l 4. Apply now Theorem 2 of the Appendix A, with F(x) a minimum polynomial for f . The choice in §1,
,
ms 2 =a 2 =log(4f)
is allowed because m may be assumed so large that the additional condition of the theorem, 0* 8*5
is likewise satisfied. Next fix the parameters ph, ah> and rji of the Theorem by Ph = ffh = *h> Since
rh =^Ze(v^
(h = 1,2,..., m).
A THEOREM BY M. CUGLANI
173
the further condition of the theorem,
also holds provided m and hence X, Hi,...,Hm are sufficiently large. There follows then from the theorem the existence of a positive constant c depending only on £ , and that of a polynomial *i rm . A(xi,...,xm) = 2 ... 2 aixl ^xi^.x^O ii=0 im=0 "' m with the following properties, (i): The coefficients &iltu9i are integers such that
and they vanish unless
(ii): AJJ ...jm(£ v> 5) vanishes for all suffixes ji ,..., ]m such that 0«Ji«nf...fO«Jm £ £h « 2 h=l
174
LECTURES ON DIOPHANTINE APPROXIMATIONS
for which the rational number Aft)
=
A^ ...lm(Ki >."vKm) = A^ um j
fe distinct from zero. Put again m
h=l r h " The choice of t implies now that
as soon as m is sufficiently large. 5. From here on the proof runs very similar to that of the case d=l of the First Approximation Theorem in Chapter 7. The slight change in notation with respect to s (which corresponds to — in the former proof) does not affect the discussion. Denote by CB, c6, and c7 three further positive constants that depend on £ , but not on m. Further let J* be the set of all systems of m integers (Ji,Jm) such that m . /* \ m li** ji^ 1*1,...,lm ^ Jm ^ rm> 2j T u ^ l ^ " 8 ) u T. • h=l T h ^ /h=l Th Then, just as in §4 of Chapter 7, Aft) * E Aj^.j^ ,...,«)\W"AW (j)eJ*
(Kitf^.dtm^™'1™,
and here TI
rm
Z... E l ji=0 i =0 m
Now the Th were chosen such that .» " 1+ ^
(h = M,...,m) .
It follows then from the construction of Hh and rn that m a x ^ ^ . . . ^ ^ " (j)eJ*
1 1 1
^
1
m
max (j)eJ* h=l
(this inequality is continued on the following page)
A THEOREM BY M. CUGIANI
175
m
Here
m
«•! (>*«!)*
g X+M '
and ^h^rh>rh, 10 *
hence
E ^^2 £ h=l T fc h=l
Therefore
and so, finally, (4):
A(1)I * (clC.)mri
(X ri ^ {S8) Hr
(m+^)  2A} .
6. We next express again
as the quotient of two integers N(i)+0 and D(i)*0 that are relatively prime. The discussion in §§67 of Chapter 7, specialised for the case d=l, may be repeated without any essential change and leads to the inequalities c6mri
and
On dividing these, it follows that
(5):
176
LECTURES ON DIOPHANTINE APPROXIMATIONS
where E* denotes the expression E* = (l7. We finally combine the upper bound (4) for  AQJ I with the lower bound (5). Then we obtain the inequality HiE
(6):
where the exponent
after a trivial simplification, may be written as E=
Or8)* " t^MJmB  ^
em 
Now .
—,,  7 =, , or £ 5aV~m~,, 0 < 0 ^ — Vm m
.
and hence
as soon as m is sufficiently large. Therefore (6) implies that ^>TE Hi ^ (CiC B C6C 7 )
,
contrary to the assumption that
X
Hi ^X = e r when m is sufficiently large. This proves the assertion. 8. It would not be difficult to extend Theorem 1 to the more general case treated in the First Approximation Theorem. There may even be a corresponding analogue of the Second Approximation Theorem; but a proof of such an analogue would perhaps require new ideas. At present it does not seem possible to replace the function e(H) by any much smaller function of H. Such an improvement would require a stronger result on the zeros of polynomials in many variables than Roth's Lemma. 9. Two simple deductions from Theorem 1 have some interest in themselves and may therefore be mentioned here. Theorem 2: Let p be a prime and q an integer such that p > q £ 2, hence (p,q) = 1.
A THEOREM BY M. CUGIANI
177
Let N = {n\l\ ir% n^,...} be a strictly increasing sequence of positive integers such that n
,/neN,
where gn is the integer nearest to (~\ . Then n
lim sup
(k+D
Proof: For every positive integer n put
"»£• *S* where dn  (Pn, gn«in) " Both dn and Pn are powers of p; Qn is divisible by qn so that n^logQn logq
, '
and it IB obvious from
1(5)° 
that
2
lim ^ = 1.
It follows that there are three positive constants yi, y2, and y3 such that 0 < Qn < YI Pn < yi pn
and hence
n
and 1^1 n ^u1 ^  1 ^u IQnlq * q * V2Qn > ° < Sn ^ VsQn
where
,^x loe/2\ u.^!iflZ
^
logp
'
iuJSU ** logp '
Here the upper bound for g^ is a consequence of the asymptotic relation
The lower bound for n in terms of Qn implies that for all sufficiently large n,
178
LECTURES ON DIOPHANTINE APPROXIMATIONS IQnlogp ^ . 91ogQn Vloglog n VlogloglogQn ' From now on let neN. By the hypothesis,
la. Qn
^ —exp ( 10nl°gP^ ^ Sn \ Vloglogn/
Kn W,
^ y Q/*9*0*0*0^^" 2^ We apply now Theorem 1, with
Since 5Vlog(4f) = SVlogT" < 9, the theorem gives
and from this, by logQnlogyi ( ^ logQn logp "* logq the assertion follows at once. 10. As a second application we construct a class of trancendental numbers which, in general, are not Liouville numbers. Theorem 3: 'Let g ^ 2 be a fixed integer, 6 a constant such that 0 < 8 < 1, {con} a» increasing infinite sequence of positive numbers tending to infinity, {j/n} a strictly increasing infinite sequence of positive integers satisfying (n = 1,2,3,...), and {an} an infinite sequence of positive integers prime to g such that
number
n=l
transcendental.
A THEOREM BY M. CUGIANI
179
Proof: Put
n=l
n=N+l
so that
The integers PN and QN are relatively prime because PN = aN +
2 an g1*"1* B &N (mod g) n=l
is prime to g. From the hypothesis,
and
Let now N be sufficiently large. Since o>n increases to infinity with n, it is obvious that (10) con Vloglogi/n
is an increasing function of n for n > N. Therefore oo
/
oo
(10) 0>n \
_ nil*" .... 1*11 ^ E g^(1+VBgBgi57 E n=N n=N Z n=N
Further n=N
n=N
1 g
"
because the integers vn are strictly Increasing with n. Hence
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LECTURES ON DIOPHANTINE APPROXIMATIONS
A
^ 1
(10) OJN logQN logg
for all sufficiently large N. Assume now that the assertion is false and that £ is algebraic, say of degree f . Then Theorem 1 may be applied with
X = 1, /i = 0, g" = g, ci = 1, ca = 1, while g1 is an arbitrary integer prime to g. But for large N, 5Vlog(4f) < (l0)a> N because CON tends to infinity. Hence it follows from the theorem that
or, what is the same,
lim sup N>~ There exist then arbitrarily large N for which
For these N,
n=N
n=N
and hence 0Pr are distinct prime ideals of K; and A.i,...,Ar are positive constants. Actually, the values of the latter do not affect the equivalence class of the pseudovaluation. One, but not both, of the numbers q and r may vanish, and if then the other number equals 1, the pseudo valuation becomes equivalent to one of the valuations that were discussed already. 5. Of particular interest are certain Non Archimedean pseudovaluations of K which are analogous to the gadic value lalg of r and may be defined similarly. For let 0 be any integral ideal of K that is distinct from both the zero ideal (0) and the unit ideal n=(l). Denote by
its factorisation into a product of powers of distinct prime ideals PI ,,,., pr, with exponents ei,...,er which are positive integers. Let further e(fij) be the order of PJ, and let pj be the rational prime that is divisible by PJ.
APPROXIMATION THEOREMS
185
Naturally the r rational primes pi,...,Pr need not all be distinct. The 0adic pseudovaluation \a g of K is now defined by the formula e(pi)logN(0) e(y r )logN(0) 6llogpl In 10  max(a (^ ..... a Irf*1*** ), where N(0) denotes the ideal norm of 9 . In the special case of the rational field r, ag becomes again the gadic value ag because Pi = (PI ),..., pr = (pr) now are principal prime ideals, and all the orders e(pi),...,e(fir) are unity. There exist elements y* 0 of K which generate principle ideals of the special form
where m and n are integral ideals that are prime to 0. Therefore
and hence e(yi)logN(8)
e(p r )logN(0)
Irl,  M It follows then that, similarly as the gadic value ag, the 0adic value has the following two properties, (I): an0 = ( a g)n for all aeK and all positive integers n; (II): aynlu = a Ij(lrl0) n for aU «€K and all rational integers n. In the special case when 0 = pe is the power of a single prime ideal, e(y)ef(y)logp
Hiai,
elogp
,
and so ag is now equivalent to the 9 adic valuation ofi. 6. The completion of K with respect to \a g, Kg say, is called the 0adic ring, and its elements are called 0adic numbers. Let similarly Kfi,...,K$r be the completions of K with respect to \a Ipi,..., \ot \?r, so that they are the piadic,..., pradic fields, respectively, which we have already considered. Just as in the gadic case, there is a onetoone correspondence A«+(oLi,...,ar) between the elements A of K$ and the sets (oti9...9ar) of one element ar in Kp , respectively. This correspondence is again preserved under addition, subtraction, and multiplication; and whenever division is possible, it is also preserved under division.
186
LECTURES ON DIOPHANTINE APPROXIMATIONS
7. In order to formulate the assertions in a convenient form, the following notation will be used. First, we put a I* = min(a , 1), a * = min( a It, 1) (i = 1,2,..., n +ra), lalr, 1),
a0, 1). Secondly, let q be any integer satisfying Then denote by ii,...,iq any system of q distinct suffixes 1, 2,...,ri+r2 arranged such that 1 * u < i2 < ... < iq * ri+r 2 ; for q=0 the system is empty. Third, «, 0f, 3" are three integral ideals distinct from (0) and o = (l) which are relatively prime in pairs. Fourth, r, rf , r11 are three nonnegative integers, and are r+r'+r" distinct prime ideals of K. It is not excluded that one or more of r, r1, rlf are zero. Fifth, Pu,..., Pi >