Randomness and quantitative aspects of the Littlewood Conjecture

In this talk, we discuss several counting problems in the theory of Diophantine approximation, specifically related to the famous Littlewood Conjecture. This conjecture has a long history, which dates back to the start of the last century. Both dynamics and lattice point counting are at the forefront of the modern techniques used to approach this and other relevant problems in the area.

By extending a well-known work of Wolfgang Schmidt, we show that in the specific context of Littlewood’s Conjecture various degrees of randomness arise. In particular, we obtain almost-everywhere counting estimates for Littlewood-type products, as well as a central limit theorem for multiplicative approximants. At the core of our work lies a new powerful equidistribution result for certain subgroups of the diagonal group on intermediate manifolds in the moduli space. This is part of a joint work with Alexander Gorodnik and Michael Bjorklund.