Kleinian Schottky groups, Patterson–Sullivan measures, and Fourier decay

Abstract

Let Γ be a Zariski-dense Kleinian Schottky subgroup of PSL2(C). Let ΛΓ⊂C be its limit set, endowed with a Patterson–Sullivan measure μ supported on ΛΓ. We show that the Fourier transform μˆ(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 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.