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.

Automated summary

This paper examines the properties of Zariski-dense Kleinian Schottky subgroups of PSL2(C), showing polynomial decay of the Fourier transform in the limit and concluding positive Fourier dimension for all limit sets. The results are an extension of Bourgain and Dyatlov's findings in PSL2(R) and rely on the decay of exponential sums based on sum-product estimates on C. The bounds on exponential sums are based on a delicate nonconcentration hypothesis proven using representation theory and regularity estimates for stationary measures of random walks on linear groups.