Fourier transforms of Gibbs measures for the Gauss map

@article{Jordan2013FourierTO,
  title={Fourier transforms of Gibbs measures for the Gauss map},
  author={Thomas Jordan and Tuomas Sahlsten},
  journal={Mathematische Annalen},
  year={2013},
  volume={364},
  pages={983-1023}
}
We investigate under which conditions a given invariant measure $$\mu $$μ for the dynamical system defined by the Gauss map $$x \mapsto 1/x \,\,{\mathrm {mod}}\,1$$x↦1/xmod1 is a Rajchman measure with polynomially decaying Fourier transform $$\begin{aligned} |\widehat{\mu }(\xi )| = O(|\xi |^{-\eta }), \quad \text {as} \quad |\xi | \rightarrow \infty . \end{aligned}$$|μ^(ξ)|=O(|ξ|-η),as|ξ|→∞.We show that this property holds for any Gibbs measure $$\mu $$μ of Hausdorff dimension greater than 1… 
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