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

Abstract: Littlewood's conjecture is a famous and long-standing open problem which states that, for every $(\alpha,\beta) \in \mathbb{R}^2$, $n\|n\alpha\|\|n\beta\|$ can be arbitrarily small for some integer $n$. This problem is closely related to the action of diagonal matrices on $\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$, and a groundbreaking result was shown by Einsiedler, Katok and Lindenstrauss from the measure rigidity for this action, saying that Littlewood's conjecture is true except on a set of Hausdorff dimension zero. In this talk, I will explain about a new quantitative result on Littlewood's conjecture which gives, for every $(\alpha,\beta) \in \mathbb{R}^2$ except on sets of small Hausdorff dimension, an estimate of the number of integers $n$ which make $n\|n\alpha\|\|n\beta\|$ small. The keys for the proof are the measure rigidity and further studies on behavior of empirical measures for the diagonal action.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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