Asymptotic Diophantine approximation: The multiplicative case

2014/07/02 by Martin Widmer, Widmer, Martin
Mathematics · #11J25 #11J54 Secondary 11H46 #37A17 #FOS: Mathematics #Number Theory (math.NT) #Primary 11J13 #math.NT #msc:11H46 #msc:11J13 #msc:11J25 #msc:11J54 #msc:37A17

paper · pdf · doi:10.48550/arxiv.1407.0427

To appear in Ramanujan Journal

arxiv created 2016/03/20 · arxiv updated 2016/03/22

Abstract

Let α and β be irrational real numbers and 0<\F<1/30. We prove a precise estimate for the number of positive integers q≤ Q that satisfy ‖qα‖⋅‖qβ‖<\F. If we choose \F as a function of Q we get asymptotics as Q gets large, provided \F Q grows quickly enough in terms of the (multiplicative) Diophantine type of (α,β), e.g., if (α,β) is a counterexample to Littlewood's conjecture then we only need that \F 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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