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On the Rational Approximations to the Powers of an Algebraic Number: Solution of Two Problems of Mahler and Mend S France

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On the Rational Approximations to the Powers of an Algebraic Number: Solution of Two Problems of Mahler and Mend S France Acta Math., 193 (2004), 175 191 (~) 2004 by Institut Mittag-Leffier. All rights reserved On the rational approximations to the powers of an algebraic number: Solution of two problems of Mahler and Mend s France by PIETRO CORVAJA and UMBERTO ZANNIER Universith di Udine Scuola Normale Superiore Udine, Italy Pisa, Italy 1. Introduction About fifty years ago Mahler [Ma] proved that if ~> 1 is rational but not an integer and if 0<l<l, then the fractional part of (~n is larger than l n except for a finite set of integers n depending on ~ and I. His proof used a p-adic version of Roth's theorem, as in previous work by Mahler and especially by Ridout. At the end of that paper Mahler pointed out that the conclusion does not hold if c~ is a suitable algebraic number, as e.g. 1 (1 + x/~ ) ; of course, a counterexample is provided by any Pisot number, i.e. a real algebraic integer c~>l all of whose conjugates different from cr have absolute value less than 1 (note that rational integers larger than 1 are Pisot numbers according to our definition). Mahler also added that "It would be of some interest to know which algebraic numbers have the same property as [the rationals in the theorem]". Now, it seems that even replacing Ridout's theorem with the modern versions of Roth's theorem, valid for several valuations and approximations in any given number field, the method of Mahler does not lead to a complete solution to his question. One of the objects of the present paper is to answer Mahler's question completely; our methods will involve a suitable version of the Schmidt subspace theorem, which may be considered as a multi-dimensional extension of the results mentioned by Roth, Mahler and Ridout. We state at once our first theorem, where as usual we denote by IIx]] the distance of the complex number x from the nearest integer in Z, i.e. Ilxll := min{Jx-mJ : mCZ}. 176 P. CORVAJA AND U. ZANNIER THEOREM 1. Let a>l be a real algebraic number and let 0<l<1. Suppose that IIanll < In for infinitely many natural numbers n. Then there is a positive integer d such that a d is a Pisot number. In particular, a is an algebraic integer. We remark that the conclusion is best possible. For assume that for some positive integer d, /3:--a d is a Pisot number. Then for every positive multiple n of d we have Ilanll << I n, where l is the dth root of the maximum absolute value of the conjugates of/3 different from/3. Here, Mahler's example with the golden ratio is typical. Also, the conclusion is not generally true without the assumption that a is algebraic; for this see the appendix. The present application of the subspace theorem seems different from previous ones, and occurs in Lemma 3 below. Related methods actually enable us to answer as well a question raised by Mend~s France about the length of the periods of the continued fractions for a n, where a is now a quadratic irrational; this appears as Problem 6 in [Me]. We shall prove, more generally, the following result: THEOREM 2. Let a>0 be a real quadratic irrational. If a is neither the square root of a rational number, nor a unit in the ring of integers of Q(a), then the period length of the continued fraction for a n tends to infinity with n. If a is the square root of a rational number, the period length of the continued fraction for a 2n+1 tends to infinity. If a is a unit, the period length of the continued fraction for a n is bounded. Clearly, if a is the square root of a rational number, then the continued frac- tion for a ~n is finite,(1) so Theorem 2 gives a complete answer to the problem of Mendbs France. The main tool in the proof of both theorems is the following new lower bound for the fractional parts of S-units in algebraic number fields. We first need a definition: Definition. We call a (complex) algebraic number a a pseudo-Pisot number if (i) lal > 1 and all its conjugates have (complex) absolute value strictly less than 1; (ii) a has an integral trace: WrQ(a)/Q(a)EZ. Of course, pseudo-Pisot numbers are "well approximated" by their trace, hence are good candidates for having a small fractional part compared to their height. The algebraic integers among the pseudo-Pisot numbers are just the usual Pisot numbers. We shall prove the following result: (1) Its length tends to infinity by a result of Pourchet, proved in greater generality in [CZ]. ON THE RATIONAL APPROXIMATIONS TO THE POWERS OF AN ALGEBRAIC NUMBER 177 MAIN THEOREM. Let FcQ • be a finitely generated multiplicative group of algebraic numbers, let 5cQ • be a non-zero algebraic number and let ~>0 be fixed. Then there are only finitely many pairs (q,u)~z• with d=[Q(u):Q] such that 1Saul>l, 5qu is not a pseudo-Pisot number and o < 115q ll < H(u)-*q -d-*. (1.1) Note again that, conversely, starting with a Pisot number a and taking q=l and u=(f ~ for n=l, 2, ... produces an infinite sequence of solutions to 0< Ilqull <H(u) -~ for a suitable s>0. The above main theorem can be viewed as a Thue-Roth inequality with "moving target", as the theorem in [CZ], where we considered quotients of power sums with in- tegral roots instead of elements of a finitely generated multiplicative group. The main application of the theorem in [CZ] also concerned continued fractions, as for our Theo- ram 2. 2. Proofs We shall use the following notation: Let K be a number field, embedded in C and Galois over Q. We denote by M/( (rasp. M~) the set of places (resp. archimedean places) of K; for each place v we denote by 1 Iv the absolute value corresponding to v, normalized with respect to K; by this we mean that if vEM~ then there exists an automorphism aEGal(K/Q) of K such that, for all xEK, Ixlv = I (x)l where d(a)=l if a(K)=KcR, and d(a)=2 otherwise (note that d(a) is now constant since K/Q is Galois). Non-archimedean absolute values are normalized accordingly, so that the product formula holds and the absolute Weit height reads H(x) = H max{l, Ixlv}. vEMK For a vector x=(xl, ...,Xn)CK n and a place vEMK we shall denote by I]xl]~ the v-norm of x, namely,(2) HxHv := max{ Ix1 Iv,..., Ixn M, (2) We believe that there will be no confusion with the previously recalled symbol Ilxll for the distance to the nearest integer. 178 P. CORVAJA AND U. ZANNIER and by H(x) the projective height H(x) = H max{lxlIv,..., Ixnl~}. vEMK We begin by proving the main theorem. First of all notice that, by enlarging if necessary the multiplicative group F, we can reduce to the situation where FcK • is the group of S-units: F= O~ = {uEK:iu[~ = 1 for all v~S} of a suitable number field K, Galois over Q, with respect to a suitable finite set of places S, containing the arehimedean ones and stable under Gatois conjugation. Our first lemma can be easily deduced from a theorem of Evertse, which in turn was obtained as an application of the already mentioned subspace theorem; we prefer to give a proof for completeness. LEMMA 1. Let K and S be as before, al, ...,an be distinct automorphisms of K, /kl,...,An be non-zero elements of K, ~>0 be a positive real number and wCS be a distinguished place. Let =~C O~ be an infinite set of solutions uEO~ of the inequality [.~lO'l (U)T... T /~n(Tn(U)lw < max{]crl (u)iw, ..., [(rn(U)[w}H(u) -~. Then there exists a non-trivial linear relation of the form alal(u)+...+anan(u)=O, aiEK, satisfied by infinitely many elements of =. Proof. Let ~ be as in the lemma. Going to an infinite subset of ~ we may assume that lal(U)lw=max{tal(U)l~, ..., lan(u)l~,} for all the involved u's. Let us consider, for each yES, n linear forms Lv,1, ..., L~,n in n variables x=(xl, ..., xn) as follows: Put L~,I (x) := Alxl +...+Anxn while for (v,i)ES• with (v,i)#(w, l), put L,,~(x)=xi. Note that the linear forms L~,I, ..., Lv,n are indeed linearly independent. Now put x = (ol(u), :.., ~d(u)) e (o~) d and consider the double product H l~ ILv,dx)lv ON THE RATIONAL APPROXIMATIONS TO THE POWERS OF AN ALGEBRAIC NUMBER 179 By multiplying and dividing by Ix~l~=l~(u)l~, and using the fact that the coordinates of x are S-units, we obtain the equality H let In~,~(x)l~ _ iAl~l(U)+...+Anan(U)lw i~(u)l;~H(x)_n. ~s~=~ Ilxll~ Since uC~, we have I~l(U)+...+ ~na~(U)lw I~I(U)I~ x < H(u)-~, and then v~s~=lII ILv,I)xllv (x)lv < The height of the point x=(al(u), ..., ad(U)) is easily compared with the height of u by the estimate H(x) ~< H(u) [~;:q] . Hence the above upper bound for the double product also gives H fl IL~"(x)lv H(x)-n-~/[K:Q]" Now, an application of the subspace theorem in the form given in e.g.
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