UW Rainwater Seminar
Winter, 2003

In a recent joint work with A. Katok and E. Lindenstrauss we proved a partial result regarding Littlewood's conjecture using methods from ergodic theory. Littlewood's conjecture states that for every two real numbers a,b the limes inferior of n for n going to infinity equals zero, where < > denotes the distance to the nearest integer. The study of measure that are invariant under two or more commuting transformations began shortly after Furstenberg's seminal paper in 1967, but a complete description of such measures has yet to be found. Partial results on this problem for multiplication by 2 and 3 on the circle group (Furstenberg's problem, or for some people conjecture) have been obtained by Lyons, Rudolph, and Johnson. I will explain how we used our result (which extends earlier work by Katok, Kalinin, Spatzier, and myself) on invariant measures on SL(3,R)/SL(3,Z) to show that the exceptions to Littlewood's conjecture form a very thin set: the Hausdorff dimension of the set of exceptions is zero.