On finite configurations in the spectra of singular measures

TL;DR

This paper proves that highly singular measures' supports necessarily contain specific finite linear patterns, leading to new dimensional bounds for measures constrained by PDEs and Fourier analysis.

Contribution

It introduces new results linking the singularity of measures to the presence of finite linear configurations in their support, improving bounds related to the $k$-wave cone.

Findings

Supports of singular measures contain prescribed finite linear patterns.

Provides new dimensional estimates for PDE- and Fourier-constrained measures.

Improves bounds related to the $k$-wave cone in certain cases.

Abstract

We establish various forms of the following certainty principle: a set $S \subset \mathbb{R}^{n}$ contains a given finite linear pattern, provided that $S$ is a support of the Fourier transform of a sufficiently singular probability measure on $\mathbb{R}^{n}$. As its main corollary, we provide new dimensional estimates for PDE- and Fourier-constrained vector measures. Those results, in certain cases of restrictions given by homogeneous operators, improve known bounds related to the notion of the $k$-wave cone.