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

Abstract

Let Γ be a Zariski-dense Kleinian Schottky subgroup of ${\mathrm{PSL}}_2(\mathbb{C})$. Let ${\mathrm{\Lambda }}_{\mathrm{\Gamma }}\subset \mathbb{C}$ be its limit set, endowed with a Patterson–Sullivan measure μ supported on ${\mathrm{\Lambda }}_{\mathrm{\Gamma }}$. We show that the Fourier transform $\hat{\mu }(\xi )$ enjoys polynomial decay as $|\xi |$ goes to infinity. As a corollary, all limit sets of Zariski-dense Kleinian groups have positive Fourier dimension. This is a ${\mathrm{PSL}}_2(\mathbb{C})$ version of the $\mathrm{PS}{L}_2(\mathbb{R})$ result of Bourgain and Dyatlov, and uses the decay of exponential sums based on Bourgain–Gamburd’s sum-product estimate on $\mathbb{C}$. 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.