Dispersion and Littlewood's conjecture

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‖<(log⁡log⁡n)3+εlog⁡n. 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.