×

On the rational approximations to the powers of an algebraic number: solution of two problems of Mahler and Mendès France. (English) Zbl 1175.11036

This interesting article is nicely described by (selected) parts of the introduction: “About fifty years ago K. Mahler [Mathematika 4, 122–124 (1957; Zbl 0208.31002)] proved that if \(\alpha> 1\) is rational but not an integer and if \(0<l<1\), then the fractional part of \(\alpha^n\) is larger than \(l^n\) except for a finite set of integers \(n\) depending on \(\alpha\) and \(l\). 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 \(\alpha\) is a suitable algebraic number, as e.g. \(\frac12 (1 + \sqrt 5)\); of course, a counterexample is provided by any Pisot number, i.e. a real algebraic integer \(\alpha>1\) all of whose conjugates different from \(\alpha\) have absolute value less than 1. 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 \(\| x\|\) the distance of the complex number \(x\) from the nearest integer in \(\mathbb Z\), i.e. \(\| x\| := \min\{| x-m| : m\in\mathbb Z\}\).
Theorem 1. Let \(\alpha>1\) be a real algebraic number and let \(0<l<1\). Suppose that \(\|\alpha^n\| < l^n\) for infinitely many natural numbers \(n\). Then there is a positive integer \(d\) such that \(\alpha^d\) is a Pisot number. In particular, \(\alpha\) is an algebraic integer.
We remark that the conclusion is best possible. For assume that for some positive integer \(d\), \(\beta:=\alpha^d\) is a Pisot number. Then for every positive multiple \(n\) of \(d\) we have \(\|\alpha^n\| \ll l^n\), where \(l\) is the \(d\)th root of the maximum absolute value of the conjugates of \(\beta\) different from \(\beta\). Here, Mahler’s example with the golden ratio is typical. Also, the conclusion is not generally true without the assumption that \(\alpha\) is algebraic.
The present application of the subspace theorem seems different from previous ones. We prove, more generally, the following result:
Theorem 2. Let \(\alpha>0\) be a real quadratic irrational. If \(\alpha\) is neither the square root of a rational number, nor a unit in the ring of integers of \(\mathbb Q(\alpha)\), then the period length of the continued fraction for \(\alpha^n\) tends to infinity with \(n\). If \(\alpha\) is the square root of a rational number, the period length of the continued fraction for \(\alpha^{2n+1}\) tends to infinity. If \(\alpha\) is a unit, the period length of the continued fraction for \(\alpha^n\) is bounded.
If \(\alpha\) is the square root of a rational number, then the continued fraction for \(\alpha^n\) is finite, so Theorem 2 gives a complete answer to a problem posed by Mendès 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.
Definition. We call a (complex) algebraic number \(\alpha\) a pseudo-Pisot number if (i) \(|\alpha | > 1\) and all its conjugates have (complex) absolute value strictly less than 1; (ii) \(\alpha\) has an integral trace: \(\text{Tr}_{\mathbb Q(\alpha)/\mathbb Q}(\alpha)\in\mathbb Z\).
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 prove the following result:
Main Theorem. Let \(\Gamma\subset\overline{\mathbb Q}^\times\) be a finitely generated multiplicative group of algebraic numbers, let \(\delta\in\overline{\mathbb Q}^\times\) be a non-zero algebraic number and let \(\varepsilon>0\) be fixed. Then there are only finitely many pairs \((q,u)\in\mathbb Z\times\Gamma\) with \(d=[\mathbb Q(u):\mathbb Q]\) such that \(|\delta qu|>1\), \(\delta qu\) is not a pseudo-Pisot number and
\[ 0 < \|\delta qu\| < H(u)^{-\varepsilon}q^{-d-\varepsilon}. \]
Note again that, conversely, starting with a Pisot number \(\alpha\) and taking \(q=1\) and \(u=\alpha^n\) for \(n=1, 2, \dots\) produces an infinite sequence of solutions to \(0< \| qu\| <H(u)^{-\varepsilon}\) for a suitable \(\varepsilon>0\).
The above main theorem can be viewed as a Thue-Roth inequality with “moving target”, as the theorem in the authors’ paper [J. Théor. Nombres Bordx. 17, No. 3, 737–748 (2005; Zbl 1159.11021)], where we considered quotients of power sums with integral roots instead of elements of a finitely generated multiplicative group. The main application of the theorem in the paper cited above also concerned continued fractions, as for our Theorem 2.”

MSC:

11J68 Approximation to algebraic numbers
11J87 Schmidt Subspace Theorem and applications
11J70 Continued fractions and generalizations
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] [CZ]Corvaja, P. & Zannier, U., On the length of the continued fraction for the ratio of two power sums. To appear inJ. Théor. Nombres Bordeaux. · Zbl 1159.11021
[2] [Ma]Mahler, K., On the fractional parts of the powers of a rational number, II.Mathematika, 4 (1957), 122–124. · Zbl 0208.31002
[3] [Me]Mendès France, M., Remarks and problems on finite and periodic continued fractions.Enseign. Math., 39 (1993), 249–257. · Zbl 0808.11007
[4] [S]Schmidt, W. M.,Diophantine Approximations and Diophantine Equations. Lecture Notes in Math., 1467. Springer, Berlin, 1991. · Zbl 0754.11020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.