Dispersion and Littlewood's conjecture

Abstract

Let $\varepsilon>0$. We construct an explicit, full-measure set of $α\in[0,1]$ such that if $γ\in \mathbb{R}$ then, for almost all $β\in[0,1]$, if $δ\in \mathbb{R}$ then there are infinitely many integers $n\geq 1$ for which \[ n \Vert nα- γ\Vert \cdot \Vert nβ- δ\Vert < \frac{(\log \log n)^{3 + \varepsilon}}{\log n}. \] 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.