Dispersion and Littlewood's conjecture

Abstract

Let $\epsilon >0$. We construct an explicit, full-measure set of $\alpha \in [0,1]$ such that if $\gamma \in \mathbb{R}$ then, for almost all $\beta \in [0,1]$, if $\delta \in \mathbb{R}$ then there are infinitely many integers $n\ge 1$ 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 $\beta$ 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.