Quantitative uniqueness properties for $L^2$ functions with fast decaying, or sparsely supported, Fourier transform

TL;DR

This paper extends quantitative uniqueness theorems to functions with rapidly decaying Fourier transforms and transfers these results to functions supported on fractal sets, broadening understanding of Fourier decay and support properties.

Contribution

It characterizes when quantitative uniqueness holds for rapidly decaying Fourier transforms and develops a transference method to apply these results to fractal support sets.

Findings

Extended Paneah-Logvinenko-Sereda theorem for fast decaying Fourier transforms.

Developed a transference result for fractal support sets.

Recovered and generalized Bourgain-Dyatlov's uniqueness results.

Abstract

This paper builds upon two key principles behind the Bourgain-Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah-Logvinenko-Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on a fractal set. As well as recovering the result of Bourgain-Dyatlov, we obtain analogous uniqueness results for denser fractals.