Kleinian Schottky groups, Patterson Sullivan measures and Fourier Decay

We show that the Fourier transform of the Patterson-Sullivan measure of a Kleinian Schottky subgroup of PSL2(C) enjoys polynomial decay. This generalises a result of Bourgain-Dyatlov for convex co-compact Fuchsian groups, and uses the decay of exponential sums based on Bourgain-Gamburd sum-product estimate on C. These bounds on exponential sums require a delicate non-concentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.