Fourier transforms of Gibbs measures for the Gauss map

We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map x↦1/xmod1 is a Rajchman measure with polynomially decaying Fourier transform

|μˆ(ξ)|=O(|ξ|−η),as|ξ|→∞.

We show that this property holds for any Gibbs measure μ 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 Queffélec–Ramaré. Our main result implies that the Fourier–Stieltjes coefficients of the Minkowski’s question mark function decay to 0 polynomially answering a question of Salem from 1943. As an application of the Davenport–Erdős–LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman–Shmerkin. 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.