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arXiv · 2108.12036

On finite configurations in the spectra of singular measures

Rami Ayoush
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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.

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Rami Ayoush. 2021-08-26. On finite configurations in the spectra of singular measures. https://arxiv.org/abs/2108.12036

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