Abstract
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),asξ→∞.
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.
μˆ(ξ)=O(ξ−η),asξ→∞.
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.
Original language  English 

Pages (fromto)  9831023 
Number of pages  41 
Journal  Mathematische Annalen 
Volume  364 
Issue number  3 
Early online date  14 Jun 2015 
DOIs  
Publication status  Published  Apr 2016 
Bibliographical note
v3: 29 pages; peerreviewed version, fixes typos and added more elaborations, and included comments on Salem's problem. To appear in Math. AnnKeywords
 42A38
 11K50
 37C30
 60F10
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Dr Thomas M Jordan
 Probability, Analysis and Dynamics
 School of Mathematics  Senior Lecturer in Pure Mathematics
 Pure Mathematics
 Ergodic theory and dynamical systems
Person: Academic , Member