An improvement of the lower bound of the number of integers in Littlewood's conjecture

In this paper, we improve the results in the author’s previous paper \cite{Usu22}, which deals with the quantitative problem on Littlewood’s conjecture. We show that, for any $0<\gamma <1$, any $(\alpha ,\beta )\in {\mathbb{R}}^2$ except on a set with Hausdorff dimension about $\sqrt{\gamma }$, any small $0<\epsilon <1$ and any large $N\in \mathbb{N}$, the number of integers $n\in [1,N]$ such that $n⟨n\alpha ⟩⟨n\beta ⟩<\epsilon$ is greater than $\gamma (\log N{)}^2\mathrm{/}(\log \log N{)}^2$ up to a universal constant.