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A randomized version of the Littlewood conjecture. (English) Zbl 1418.11108

Summary: The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering \(\mathbb{R}^2\) by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each hyperbola by a random amount which depends on the denominator of the corresponding rational. Using a randomized covering argument we prove that, not only is this randomized version of the Littlewood Conjecture true for almost all choices of centers, an even stronger statement with an extra factor of a logarithm also holds.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
60D05 Geometric probability and stochastic geometry
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References:

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