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

Let Gamma be a Zariski-dense Kleinian Schottky subgroup of PSL2(C). Let Lambda(Gamma) subset of C be its limit set, endowed with a Patterson-Sullivan measure mu supported on Lambda(Gamma). We show that the Fourier transform b (mu) over cap(xi) enjoys polynomial decay as vertical bar xi vertical bar goes to infinity. As a corollary, all limit sets of Zariski-dense Kleinian groups have positive Fourier dimension. This is a PSL2(C) version of the PSL2(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 nonconcentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.