Fractal uncertainty principle for discrete Cantor sets with random 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 $\log A/\log M\in (0,2/3)$. We prove that the FUP with an improved exponent over Dyatlov-Jin $\href{https://doi.org/10.48550/arXiv.2107.08276}{\textrm{DJ-1}}$ holds for almost all alphabets, asymptotically as $M\to\infty$. 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.