Invariant measures and the set of exceptions to Littlewood’s conjecture

Pages 513-560 from Volume 164 (2006), Issue 2 by Manfred Einsiedler, Anatole Katok, Elon Lindenstrauss

Abstract

We classify the measures on $SL(k,\mathbb{R}) / SL(k,\mathbb{Z})$ which are invariant and ergodic under the action of the group $A$ of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood’s conjecture has Hausdorff dimension zero.

DOI
https://doi.org/10.4007/annals.2006.164.513
MR
2247967
zbMATH
1109.22004
Milestones
Received: 5 September 2003
Revised: 25 January 2005
Accepted: 27 September 2004
Published online: 1 March 2009

Authors

Manfred Einsiedler

Department of Mathematics, Princeton University, Princeton, NJ 08544, United States

Anatole Katok

Department of Mathematics, The Pennsylvania State University, State College, PA 16802, United States

Elon Lindenstrauss

Department of Mathematics, Princeton University, Princeton, NJ 08544, United States