KLEINIAN SCHOTTKY GROUPS, PATTERSON-SULLIVAN MEASURES, AND FOURIER DECAY

摘要

Let Γ be a Zariski-dense Kleinian Schottky subgroup of PSL_2(C).Let Λ_Γ ? C be its limit set, endowed with a Patterson-Sullivan measure μ supported on ΛΓ.We show that the Fourier transform μ(ξ) enjoys polynomial decay as |ξ| goes to infinity.As a corollary, all limit sets of Zariski-dense Kleinian groups have positive Fourier dimension.This is a PSL_2(C) version of the PSL_2(R) result of Bourgain and Dyatlov, and uses the decay of exponential sums based on Bourgain-Gamburd's sum-product estimate on C.These bounds on exponential sums require a delicate nonconcen-tration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.