ETH Zurich

### The set of exceptions to the inhomogeneous uniform version of Littlewood conjecture

Ergodic Theory and Dynamical Systems Seminar

8th February 2024, 2:00 pm – 3:00 pm

Fry Building,
G.07

We say that a pair of real numbers (a,b) satisfies the inhomogeneous uniform version of Littlewood conjecture if liminf_{q\to \infty} q=0 for any pair of real numbers (c,d), where <⋅> denotes the distance to the nearest integer. In 2011 Shapria showed that the set of exceptions to the inhomogeneous Littlewood conjecture has Lebesgue measure zero. In this talk, we compute that the Hausdorff dimension of the set of exceptions is equal to 1. I plan to outline the several ingredients of the proof: Dani’s correspondence in the space of affine lattices, a measure classification theorem of Einsiedler and Lindenstrauss, and estimates on the escape of mass and entropy for higher rank diagonal actions.

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