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Zbl 1194.11075
Venkatesh, Akshay
The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture. (English)
[J] Bull. Am. Math. Soc., New Ser. 45, No. 1, 117-134 (2008). ISSN 0273-0979; ISSN 1088-9485/e

This document is a brief report on the work of {\it M. Einsiedler, A. Katok} and {\it E. Lindenstrauss} on the Littlewood conjecture [Ann. Math. (2) 164, No. 2, 513--560 (2006; Zbl 1109.22004)]. For $x\in\Bbb R$, let $\Vert x\Vert$ denote the distance from $x$ to the nearest integer. The Littlewood conjecture asserts that $$\liminf_{n\ge 1}\,n\Vert n\alpha\Vert\ \Vert n\beta\Vert= 0,\tag i$$ 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.\par This result is proved using ideas from dynamics: namely, by studying the action of coordinate dilations on the space of lattices in $\Bbb R^3$. 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.\par 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 $\Bbb R/\Bbb Z$; (iii) A sketch of idea to study the picture transeverse to the acting group.
[Ranjeet Sehmi (Chandigarh)]

MSC 2000:: *11J13 Simultaneous homogeneous approximation, linear forms
37A35 Invariants of ergodic theory
11H46 Products of linear forms
37A45 Relations of ergodic theory with number theory and harmonic analysis

Keywords: Littlewood conjecture; Oppenheim conjecture; coordinate dilations

Citations: Zbl 1109.22004

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