On a lower bound of the number of integers in Littlewood's conjecture

TL;DR

This paper provides a quantitative lower bound on the number of integers satisfying Littlewood's conjecture for most pairs of real numbers, linking measure behavior on a homogeneous space to number-theoretic properties.

Contribution

It introduces a new quantitative approach to Littlewood's conjecture by analyzing empirical measures on a homogeneous space, estimating the size of the exceptional set.

Findings

For most pairs, the count exceeds a logarithmic bound.

The exceptional set has Hausdorff dimension close to zero.

The method relates measure dynamics to number-theoretic properties.

Abstract

We show that, for any $0<\gamma<1/2$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrt{\gamma}$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such that $n\langle n\alpha\rangle\langle n\beta\rangle<\varepsilon$ is greater than $\gamma\varepsilon\log N$ up to a uniform constant. This can be seen as a quantitative result on the fact that the exceptional set to Littlewood's conjecture has Hausdorff dimension zero, obtained by M. Einsiedler, A. Katok and E. Lindenstrauss in 2000's. For the proof, we study the behavior of the empirical measures with respect to the diagonal action on $\rm{SL}(3,\mathbb{R})/\rm{SL}(3,\mathbb{Z})$ and show that we can obtain a quantitative result on Littlewood's conjecture for $(\alpha,\beta)$ if the corresponding empirical measures are well-behaved. We also estimate Hausdorff dimension of the exceptional set to be small.