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On the Littlewood conjecture in fields of power series. (English) Zbl 1223.11083

Akiyama, Shigeki (ed.) et al., Probability and number theory – Kanazawa 2005. Proceedings of the international conference on probability and number theory, Kanazawa, Japan, June 20–24, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-43-3/hbk). Adv. Stud. Pure Math. 49, 1-20 (2007).
From the text: A famous problem in simultaneous Diophantine approximation is the Littlewood conjecture. It claims that, for any given pair \((\alpha,\beta)\) of real numbers, we have \[ \inf_{q\geq 1} q \cdot\| q\alpha\| \cdot \| q\beta\| = 0, \tag{1.1} \] where \(\|\cdot\|\) denotes the distance to the nearest integer. Despite some recent remarkable progress [A. D. Pollington and S. Velani, Acta Math. 185, 287–306 (2000; Zbl 0970.11026), M. Einsiedler, A. Katok and E. Lindenstrauss, Ann. Math. (2) 164, No. 2, 513–560 (2006; Zbl 1109.22004)], this remains an open problem.
The present note is devoted to the analogous question in fields of power series. Given an arbitrary field \(\mathbf k\) and an indeterminate \(X\), we define a norm \(|\cdot|\) on the field \(\mathbf k((X^{-1}))\) by setting \(| 0| = 0\) and, for any non-zero power series \(F = F(X) = \sum^{+\infty}_{h=-m} f_hX^{-h}\) with \(f_{-m} \neq 0\), by setting \(| F| = 2^m\). We write \(\| F\|\) to denote the norm of the fractional part of \(F\), that is, of the part of the series which comprises only the negative powers of \(X\). In analogy with (1.1), we ask whether we have \[ \inf_{q\in\mathbf k[X]\backslash\{0\}} | q|\cdot\| q\Theta\| \cdot \| q\Phi\| = 0 \tag{1.2} \] for any given \(\Theta\) and \(\Phi\) in \(\mathbf k((X^{-1}))\). A negative answer to this question has been obtained by H. Davenport and D. J. Lewis [Mich. Math. J. 10, 157–160 (1963; Zbl 0107.04202)] when the field \(\mathbf k\) is infinite. As far as we are aware, the problem is still unsolved when \(\mathbf k\) is a finite field (the papers by J. V. Armitage [Mathematika 16, Part 1, No. 31, 101–105 (1969); corrigendum and addendum 17, 173–178 (1970; Zbl 0188.35002)], dealing with finite fields of characteristic greater than or equal to 5, are erroneous, as kindly pointed out to us by Bernard de Mathan).
A first natural question regarding this problem can be stated as follows:
Question 1. Given a badly approximable power series \(\Theta\), does there exist a power series \(\Phi\) such that the pair \((\Theta,\Phi)\) satisfies non-trivially the Littlewood conjecture?
First, we need to explain what is meant by non-trivially and why we restrict our attention to badly approximable power series, that is, to power series from the set \[ \mathbf{Bad} = \{ \Theta\in\mathbf k((X^{-1})) : \inf_{q\in\mathbf k[X]\backslash\{0} | q|\cdot\| q\Theta\| > 0\}. \] Obviously, (1.2) holds as soon as \(\Theta\) or \(\Phi\) does not belong to \(\mathbf{Bad}\). This is also the case when 1, \(\Theta\) and \(\Phi\) are linearly dependent over \(\mathbf k[X]\). Hence, by non-trivially, we simply mean that both of these cases are excluded.
In the present paper, we answer positively Question 1 by using the constructive approach developed in our article [J. Lond. Math. Soc., II. Ser. 73, No. 2, 355–366 (2006; Zbl 1093.11052)]. Our method rests on the basic theory of continued fractions and works without any restriction on the field \(\mathbf k\). Actually, our result is much more precise and motivates the investigation of a stronger question, introduced and discussed in Section 2. Section 3 is concerned with the Littlewood conjecture for pairs of algebraic power series. When \(\mathbf k\) is a finite field, we provide several examples of such pairs for which (1.2) holds. In particular, we show that there exist infinitely many pairs of quartic power series in \(\mathbb F_3((X^{-1}))\) that satisfy non-trivially the Littlewood conjecture. It seems that no such pair was previously known.
For the entire collection see [Zbl 1132.11001].

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11J61 Approximation in non-Archimedean valuations
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