@article{jordan_fourier_2015,
	title = {Fourier transforms of {Gibbs} measures for the {Gauss} map},
	url = {http://arxiv.org/abs/1312.3619},
	abstract = {We investigate under which conditions a given invariant measure µ for the dynamical system defined by the Gauss map x → 1/x mod 1 is a Rajchman measure with polynomially decaying Fourier transform {\textbar}µ(ξ){\textbar} = O({\textbar}ξ{\textbar}−η), as {\textbar}ξ{\textbar} → ∞. We show that this property holds for any Gibbs measure µ of Hausdorff dimension at least 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 at least 1/2 on badly approximable numbers, which extends the constructions of Kaufman and Queff´elec-Ramar´e. As an application of the Davenport-Erd˝os-LeVeque criterion on equidistribution, we obtain that µ almost every number is normal in all bases extending in part a recent result of Hochman-Shmerkin to infinite alphabets. 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.},
	language = {en},
	urldate = {2020-12-14},
	journal = {arXiv:1312.3619 [math]},
	author = {Jordan, Thomas and Sahlsten, Tuomas},
	month = feb,
	year = {2015},
	note = {arXiv: 1312.3619},
	keywords = {42A38 (Primary), 11K50, 37C30, 60F10 (Secondary), Mathematics - Classical Analysis and ODEs, Mathematics - Dynamical Systems, Mathematics - Number Theory},
}

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