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Let $\Gamma$ be a Zariski dense Kleinian Schottky subgroup of PSL2(C). Let$\Lambda(\Gamma)$ be its limit set, endowed with a Patterson-Sullivan measure$\mu$ supported on $\Lambda(\Gamma)$. We show that the Fourier transform$\widehat{\mu}(\xi)$ enjoys polynomial decay as $\vert \xi \vert$ goes toinfinity. This is a PSL2(C) version of the result of Bourgain-Dyatlov [8], anduses the decay of exponential sums based on Bourgain-Gamburd sum-productestimate on C. These bounds on exponential sums require a delicatenon-concentration hypothesis which is proved using some representation theoryand regularity estimates for stationary measures of certain random walks onlinear groups.

Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay