On construction of bounded sets not admitting a general type of Riesz spectrum

TL;DR

This paper constructs specific bounded sets in real space that do not admit Riesz spectra containing certain structured sets, advancing understanding of the limitations of exponential bases in harmonic analysis.

Contribution

It introduces new constructions of sets lacking Riesz spectra with periodic or arithmetic progression structures, extending previous techniques in spectral set theory.

Findings

Constructed sets exclude Riesz spectra with periodic structures

Demonstrated sets with no Riesz spectrum containing long arithmetic progressions

Showed existence of small measure sets with non-frame exponential systems

Abstract

Despite the recent advances in the theory of exponential Riesz bases, it is yet unknown whether there exists a set $S \subset \mathbb{R}^d$ which does not admit a Riesz spectrum, meaning that for every $\Lambda \subset \mathbb{R}^d$ the set of exponentials $e^{2\pi i \lambda \cdot x}$ with $\lambda\in\Lambda$ is not a Riesz basis for $L^2(S)$. As a meaningful step towards finding such a set, we construct a set $S \subset [-\frac{1}{2}, \frac{1}{2}]$ which does not admit a Riesz spectrum containing a nonempty periodic set with period belonging in $\alpha \mathbb{Q}_+$ for any fixed constant $\alpha > 0$, where $\mathbb{Q}_+$ denotes the set of all positive rational numbers. In fact, we prove a slightly more general statement that the set $S$ does not admit a Riesz spectrum containing arbitrarily long arithmetic progressions with a fixed common difference belonging in $\alpha \mathbb{N}$. Moreover, we show that given any countable family of separated sets $\Lambda_1, \Lambda_2, \ldots \subset \mathbb{R}$ with positive upper Beurling density, one can construct a set $S \subset [-\frac{1}{2}, \frac{1}{2}]$ which does not admit the sets $\Lambda_1, \Lambda_2, \ldots$ as Riesz spectrum. An interesting consequence of our results is the following statement. There is a set $V \subset [-\frac{1}{2}, \frac{1}{2}]$ with arbitrarily small Lebesgue measure such that for any $N \in \mathbb{N}$ and any proper subset $I$ of $\{ 0, \ldots, N-1 \}$, the set of exponentials $e^{2\pi i k x}$ with $k \in \cup_{n \in I} (N\mathbb{Z} {+} n)$ is not a frame for $L^2(V)$. The results are based on the proof technique of Olevskii and Ulanovskii in 2008.