Fourier Frames on Salem Measures

TL;DR

This paper constructs various $s$-dimensional Salem measures that lack Fourier frames, revealing limitations of Fourier analysis on these fractal measures and exploring the conditions under which Fourier frames can or cannot exist.

Contribution

It introduces new examples of Salem measures without Fourier frames, including random and deterministic constructions, and develops methods to prove nonexistence of Fourier frames for these measures.

Findings

Salem measures in the unit interval can lack Fourier frames.

Certain planar Salem measures can have orthonormal bases of exponentials.

The methods suggest potential extensions to higher dimensions.

Abstract

For every $0<s\leq 1$ we construct $s$-dimensional Salem measures in the unit interval that do not admit any Fourier frame. Our examples are generic for each $s$, including all existing types of Salem measures in the literature: random Cantor sets (convolutions, non-convolutions), random images, and deterministic constructions on Diophantine approximations. They even appear almost surely as Brownian images. We also develop different approaches to prove the nonexistence of Fourier frames on different constructions. Both the criteria and ideas behind the constructions are expected to work in higher dimensions. On the other hand, we observe that a weighted arc in the plane can be a $1$-dimensional Salem measure with orthonormal basis of exponentials. This leaves whether there exist Salem measures in the real line with Fourier frames or even orthonormal basis of exponentials a subtle problem.