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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