Decay of Fourier coefficients for Furstenberg measures

TL;DR

This paper proves that Furstenberg measures associated with certain probability measures on SL_2(R) have Fourier coefficients that vanish at infinity, indicating they are Rajchman measures, thus advancing understanding of their harmonic analysis properties.

Contribution

It establishes that Furstenberg measures with finite second moments are Rajchman measures, improving previous results by Jialun Li.

Findings

Fourier coefficients of Furstenberg measures tend to zero at infinity

Furstenberg measures are Rajchman measures under finite second moment condition

Enhances understanding of harmonic analysis of Furstenberg measures

Abstract

Let $\nu$ be the Furstenberg measure associated with a non-elementary probability measure $\mu$ on SL_2(R). We show that, when $\mu$ has a finite second moment, the Fourier coefficients of $\nu$ tend to zero at infinity. In other words, $\nu$ is a Rajchman measure. This improves a recent result of Jialun Li.