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

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

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Wenyu Pan, Penn State University
Fine Hall 801

Let $\Gamma$ be a Zariski dense Kleinian Schottky subgroup of $\mathrm{PSL}_2(\mathbb{C})$. Let $\Lambda(\Gamma)\subset \mathbb{C}$ 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 to infinity. This is a $\mathrm{PSL}_2(\mathbb{C})$ version of the result of Bourgain-Dyatlov, and uses the decay of exponential sums based on Bourgain-Gamburd sum-product estimate on $\mathbb{C}$. 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 random walks on $\mathrm{SL}_2(\mathbb{C})$. This is a work on progress with Frédéric Naud and Jialun Li.