  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.

Classical Analysis and ODEs