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On the Littlewood conjecture in simultaneous Diophantine approximation. (English) Zbl 1093.11052

Suppose that the function \( \phi \) is positive and nonincreasing over the positive integers and that \( \phi (1) = 1 \) while both \( \lim \phi (q) = 0 \) and \( \lim q. \phi (q) = \inf \) as \( q \rightarrow \inf \). \( \| . \| \) denotes distance to the nearest integer. We are concerned with a fixed real number \( \alpha \) such that for \( q \geq 1 \), \[ \inf \{ q . \| q . \alpha \| \} > 0.\tag \(*\) \] It is shown that there exists a \( \beta \), also having the property \((*)\) attributed to \( \alpha \), for which \[ q . \| q \alpha \| . \| q \beta \| \leq 1/\{q . \phi (q) \}, \tag{\(**\)} \] for infinitely many integers \( q \). In particular \[ \inf \{ q . \| q \alpha \| . \| q \beta \| \} = 0 , \] for the \( \alpha \) and \( \beta \). The set of all such \( \beta \) can be effectively constructed. A related result, namely that there exists a \( \beta \) also having property \((*)\) and such that \[ q . \| q \alpha \| . \| q \beta \| \leq 1/\log(q), \] for infinitely many positive \( q \) has been given by A. D. Pollington and S. Velani [Acta Math. 185, 287–306 (2000; Zbl 0970.11026)] who also establish that the set of all such \( \beta \) is of Hausdorff dimension one. However the authors of the paper under review make use of regular continued fraction theory alone. Their treatment is elementary but far from being simple.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11J70 Continued fractions and generalizations

Citations:

Zbl 0970.11026
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