Analysis and PDE Seminar: Fourier dimension for limit sets

Seminar | January 29 | 4-5 p.m. | 740 Evans Hall

 Semyon Dyatlov, University of California, Berkeley

 Department of Mathematics

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.