Quantitative Fourier decay for Patterson-Sullivan measures of dimension larger than $1/2$

Félix Lequen, Tuomas Sahlsten

arXiv:2607.18010v1 Primary: math.DS Secondary: math.CA, math.MG

Abstract

We give an elementary proof of power Fourier decay for Patterson-Sullivan measures associated to convex co-compact Schottky groups of dimension $δ>\frac12$, obtaining the explicit decay exponent $\frac{δ(2δ- 1)}{(2δ+ 1)(3 - δ)}$. The proof replaces the technical machinery used in previous works, such as sum-product estimates, renewal theory, Dolgopyat methods, and $L^2$ flattening, by elementary oscillatory integral estimates, hyperbolic geometry, and a duality argument based on the transpose Schottky group.

Disclosure

“Acknowledgements We thank Semyon Dyatlov, Thomas Jordan, Gaétan Leclerc, Alexandre Minetto and Tomas Persson for useful discussions. AI statement AI tools were used for proofreading the paper. References [1] A. Algom, Y. Chang, M. Wu, Y.-L. Wu. Van der Corput and metric theorems for geometric progressions for self-similar measures. Math. Ann”
PDF page 17

Classification: Proofreading, grammar, or spelling
Multiplier: 1
Verified: 2026-08-23T19:01:39.978331Z

Structural counts

Pages 19 pdf

Theorems 1 source

Lemmas 18 source

Propositions 0 source

Corollaries 0 source

Definitions 1 source

Displayed equations 122 source

Bibliography entries 42 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file PS.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.