Fractal uncertainty principle for random Cantor sets

TL;DR

This paper extends the fractal uncertainty principle to new random Cantor sets in real space, demonstrating that with high probability, a specific Fourier decay rate holds, based on a novel random construction.

Contribution

It introduces a different random construction of Cantor sets in R and proves the FUP with a certain exponent, supported by Fourier decay estimates and concentration of measure.

Findings

FUP with exponent >=1/2-3d/4- holds for new random Cantor sets in R

Fourier decay estimates are established for the measures of these sets

High probability results are achieved via concentration of measure phenomena

Abstract

We continue our investigation of the fractal uncertainty principle (FUP) for random fractal sets. In the prequel (arXiv:2107.08276), we considered the Cantor sets in the discrete setting with alphabets randomly chosen from a base of digits so the dimension d is in (0,2/3). We proved that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4- holds for these discrete Cantor sets with random alphabets. In this sequel, we construct random Cantor sets with dimension d in (0,2/3) in R via a different random procedure from the one in the prequel. We prove that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4- holds. The proof follows from establishing a Fourier decay estimate of the corresponding random Cantor measures, which is in turn based on a concentration of measure phenomenon in an appropriate probability space for the random Cantor sets.