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

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.

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