A class of random Cantor measures, with applications

TL;DR

This paper surveys recent results on the geometry of spatially independent martingales, focusing on fractal measures, their Fourier decay, and applications to harmonic analysis and arithmetic structure.

Contribution

It introduces a general class of random Cantor measures with simplified proofs and explores their Fourier and convolution properties for various applications.

Findings

Analysis of Fourier decay of the measures

Results on self-convolutions of the measures

Applications to the restriction problem and arithmetic structure

Abstract

We survey some of our recent results on the geometry of spatially independent martingales, in a more concrete setting that allows for shorter, direct proofs, yet is general enough for several applications and contains the well-known fractal percolation measure. We study self-convolutions and Fourier decay of measures in our class, and present applications of these results to the restriction problem for fractal measures, and the connection between arithmetic structure and Fourier decay.