Abstract
Let \(\alpha \) and \(\beta \) be irrational real numbers and \(0<\varepsilon <1/30\). We prove a precise estimate for the number of positive integers \(q\le Q\) that satisfy \(\Vert q\alpha \Vert \cdot \Vert q\beta \Vert <\varepsilon \). If we choose \(\varepsilon \) as a function of Q, we get asymptotics as Q gets large, provided \(\varepsilon Q\) grows quickly enough in terms of the (multiplicative) Diophantine type of \((\alpha ,\beta )\), e.g., if \((\alpha ,\beta )\) is a counterexample to Littlewood’s conjecture, then we only need that \(\varepsilon Q\) tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts and sheds some light on a recent question of Lê and Vaaler.
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Notes
By non-increasing we mean that \(x,y \in [1,\infty )\) and \(x\le y\) implies that \(\phi (x)\ge \phi (y)\).
With respect to the Lebesgue measure.
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Acknowledgments
This article was initiated during a visit at the University of York, and, in parts, motivated by a question of Sanju Velani. I am very grateful to Victor Beresnevich, Alan Haynes and Sanju Velani for many interesting and stimulating discussions and their encouragement. I also would like to thank Thái Hoàng Lê for fruitful discussions, drawing my attention to [2], and for pointing out an error in an early draft of the manuscript.
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Widmer, M. Asymptotic diophantine approximation: the multiplicative case. Ramanujan J 43, 83–93 (2017). https://doi.org/10.1007/s11139-016-9779-z
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DOI: https://doi.org/10.1007/s11139-016-9779-z