Seminar

For d > k ≥ 1 consider an ergodic R^d-action on a probability space, and take a smooth k-dimensional submanifold M of R^d. When can one prove an equidistribution theorem for averages over dilated copies of M? A well-studied special case is that of spherical averages. Recently we (jointly with Prasuna Bandi and Reynold Fregoli) proved a quantitative theorem of this kind assuming effective multiple mixing of the action. This works for diagonal actions on homogeneous spaces, and as a special case applies to studying a new uniform version of Littlewood’s Conjecture. I'll probably skip the application though, and instead would like to discuss some open questions.