
Mathematics 2013
Fourier transforms of Gibbs measures for the Gauss mapAbstract: We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$\widehat{\mu}(\xi) = O(\xi^{\eta}), \quad \text{as } \xi \to \infty.$$ We show that this property holds for any Gibbs measure $\mu$ of Hausdorff dimension greater than $1/2$ with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than $1/2$ on badly approximable numbers, which extends the constructions of Kaufman and Queff\'elecRamar\'e. Our main result implies that the FourierStieltjes coefficients of the Minkowski's question mark function decay to $0$ polynomially answering a question of Salem from 1943. As an application of the DavenportErd\H{o}sLeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by HochmanShmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents.
