Speaker
Emilio Corso, Pennsylvania State Univeristy
Abstract
“In the late sixties, Furstenberg formulated a series of dynamical and geometric rigidity conjectures aimed at capturing the heuristic principle that expansions of real numbers in multiplicatively independent integer bases are uncorrelated. Among these features the intersection conjecture, concerning the intersection of two closed sets invariant under multiplication by independent integers. The conjecture was settled separately, and with vastly different methods, by Shmerkin and Wu half a century after its formulation, with a further streamlined argument given subsequently by Austin. In joint work with Shmerkin, we provide a higher-rank extension of the result, which handles intersections of any finite collection of invariant sets and, more generally, slices of products of such sets with arbitrary affine subspaces.
As in Shmerkin’s approach for the rank-one case, the argument hinges upon a precise understanding of Frostman exponents, via L^q spectra, for certain dynamically defined self-similar measures. The relevant dimensional formula, itself a manifestation of rigidity in this context, is in turn established via an L^q-flattening result for norms of discretized convolutions, pertaining to the realm of additive combinatorics and akin in spirit to the one of Khalil, which will be amply discussed in the conference as conducive to quantitative Fourier decay rates and exponential rates of mixing.”