Activities This Week

I will discuss some of the interactions between number theory and the spectral theory of the Laplacian. Some have very classical background, such as the connection with lattice point problems. Others are newer, including connections with Random Matrix Theory, the zeros of the Riemann zeta function, and the work of Maryam Mirzakhani on the moduli space of hyperbolic surfaces.

Kaufman (1984) and later Mosquera-Shmerkin (2018) showed that Bernoulli convolutions exhibit fast Fourier decay when perturbed by a smooth non-linear map. This is remarkable, since by a classical Theorem of Erdos (1939) many Bernoulli convolutions don’t have Fourier decay at all. We will present an extension of this result to all self-similar measures: Any smooth non-linear perturbation of a self-similar measure enjoys fast (polynomial) Fourier decay. Joint with Yuanyang Chang, Meng Wu, and Yu-Liang Wu.