zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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, let x denote the distance from x to the nearest integer. The Littlewood conjecture asserts that

lim inf n1 nnαnβ=0,(i)

whatever be α, β. M. Einsiedler, A. Katok and E. Lindenstrauss have proved that the set of α, β 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 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 x2x, x3x on /; (iii) A sketch of idea to study the picture transeverse to the acting group.

MSC:
11J13Simultaneous homogeneous approximation, linear forms
37A35Entropy and other invariants, isomorphism, classification (ergodic theory)
11H46Products of linear forms
37A45Relations of ergodic theory with number theory and harmonic analysis