Abstract
Let ε>0. We construct an explicit, full-measure set of α∈[0,1] such that if γ∈R then, for almost all β∈[0,1], if δ∈R then there are infinitely many integers n⩾1 for whichn‖nα−γ‖⋅‖nβ−δ‖<(loglogn)3+εlogn. This is a significant quantitative improvement over a result of the first author and Zafeiropoulos. We show, moreover, that the exceptional set of β has Fourier dimension zero, alongside further applications to badly approximable numbers and to lacunary diophantine approximation. Our method relies on a dispersion estimate and the Three Distance Theorem.
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