The Analysis and PDE seminar will take place Monday, January 29, in 740 Evans from 4:10 to 5 pm.

Title: Fourier dimension for limit sets

Abstract: For a finite measure $\mu$ on the real line, its Fourier dimension is defined using the rate of polynomial decay of the Fourier transform $\hat{\mu }$. The Fourier dimension of $\mu$ may be much smaller than the Hausdorff dimension of the support of $\mu$: a classical example is the Cantor measure on the mid-third Cantor set which has Fourier dimension equal to 0.

I will present a joint result with J. Bourgain showing that the Patterson-Sullivan measure on the limit set of a convex co-compact group of fractional linear transformations has positive Fourier dimension. The proof uses advanced tools from additive combinatorics (the discretized sum-product theorem) and exploits the fact that fractional linear transformations are (generally) not linear. An application is a new spectral gap result for convex co-compact hyperbolic surfaces.