Dispersion and Littlewood's conjecture

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 $$n\mathrm{\parallel }n\alpha -\gamma \mathrm{\parallel }\cdot \mathrm{\parallel }n\beta -\delta \mathrm{\parallel }<\frac{(\log \log n{)}^{3+\epsilon }}{\log n}\mathrm{.}$$ 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.