Bounds on the fractal uncertainty exponent and a spectral gap

TL;DR

This paper investigates the fractal uncertainty principle for discrete Cantor sets, showing how the FUP exponent varies with the alphabet's dimension and providing bounds that relate to spectral gaps in quantum maps.

Contribution

It presents new bounds on the FUP exponent for Cantor sets with large alphabets and links these results to spectral gaps in open quantum baker's maps.

Findings

FUP exponent can be exponentially small for certain large alphabet Cantor sets.

For smaller dimensions, the FUP exponent can be made arbitrarily close to the optimal bound under Diophantine conditions.

Application to spectral gaps in open quantum baker's maps demonstrates the practical relevance.

Abstract

We prove two results on Fractal Uncertainty Principle (FUP) for discrete Cantor sets with large alphabets. First, we give an example of an alphabet with dimension $\delta \in (\frac12,1)$ where the FUP exponent is exponentially small as the size of the alphabet grows. Secondly, for $\delta \in (0,\frac12]$ we show that a similar alphabet has a large FUP exponent, arbitrarily close to the optimal upper bound of $\frac12-\frac\delta2$, if we dilate the Fourier transform by a factor satisfying a generic Diophantine condition. We give an application of the latter result to spectral gaps for open quantum baker's maps.