# 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