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

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.