Fractal uncertainty principle for discrete Cantor sets with random alphabets

TL;DR

This paper proves an improved fractal uncertainty principle for discrete Cantor sets with random alphabets, showing it holds for almost all cases as the base size increases, and achieves optimal exponents under certain conditions.

Contribution

It establishes the FUP with an improved exponent for almost all random alphabets in discrete Cantor sets, extending previous results and identifying optimal conditions.

Findings

FUP holds for almost all alphabets as M increases

Achieves the best possible exponent under Fourier decay or additive energy assumptions

Uses concentration of measure in the space of alphabets

Abstract

In this paper, we investigate the fractal uncertainty principle (FUP) for discrete Cantor sets, which are determined by an alphabet from a base of digits. Consider the base of M digits and the alphabets of cardinality A such that all the corresponding Cantor sets have a fixed dimension 0<log A/log M<2/3. We prove that the FUP with an improved exponent over Dyatlov-Jin (arXiv:1608.02238) holds for almost all alphabets, asymptotically as M tends to infinity. Our result provides the best possible exponent when the Cantor sets enjoy either the strongest Fourier decay assumption or strongest additive energy assumption. The proof is based on a concentration of measure phenomenon in the space of alphabets.