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The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture. (English) Zbl 1194.11075

This document is a brief report on the work of M. Einsiedler, A. Katok and E. Lindenstrauss on the Littlewood conjecture [Ann. Math. (2) 164, No. 2, 513–560 (2006; Zbl 1109.22004)]. For $x\in ℝ$, let $\parallel x\parallel$ denote the distance from $x$ to the nearest integer. The Littlewood conjecture asserts that

$\underset{n\ge 1}{lim inf}\phantom{\rule{0.166667em}{0ex}}n\parallel n\alpha \parallel \phantom{\rule{4pt}{0ex}}\parallel n\beta \parallel =0,\phantom{\rule{2.em}{0ex}}\left(\mathrm{i}\right)$

whatever be $\alpha$, $\beta$. M. Einsiedler, A. Katok and E. Lindenstrauss have proved that the set of $\alpha$, $\beta$ for which (i) fails has Hausdorff dimension 0. The aim of this article is to discuss and give some of the context around this result.

This result is proved using ideas from dynamics: namely, by studying the action of coordinate dilations on the space of lattices in ${ℝ}^{3}$. These ideas build on the work of many e.g. Katok-Spatzier, Kalinin-Spatzier, Einsiedler-Katok and Lindenstrauss etc. The result is important because of the techniques and results in dynamics that enter into its proof.

The following points have been stressed in this article: (i) Dynamics arises from a symmetry group and the historical context of this type of connections; (ii) dynamics that is needed is similar to the simultaneous actions of $x\to 2x$, $x\to 3x$ on $ℝ/ℤ$; (iii) A sketch of idea to study the picture transeverse to the acting group.

##### MSC:
 11J13 Simultaneous homogeneous approximation, linear forms 37A35 Entropy and other invariants, isomorphism, classification (ergodic theory) 11H46 Products of linear forms 37A45 Relations of ergodic theory with number theory and harmonic analysis