  Given a set $\mathcal{S}$ of positive measure on the circle and a set of
integers $\Lambda$, one may consider the family of exponentials
$E\left(\Lambda\right):=\left\{ e^{i\lambda t}\right\}_{\lambda\in\Lambda}$ and
ask whether it is a Riesz sequence in the space
$L^{2}\left(\mathcal{S}\right)$. We focus on this question in connection with
some arithmetic properties of the set of frequencies. Improving a result of
Bownik and Speegle, we construct a set $\mathcal{S}$ such that
$E\left(\Lambda\right)$ is never a Riesz sequence if $\Lambda$ contains
arbitrary long arithmetic progressions of length $N$ and step
$\ell=O\left(N^{1-\varepsilon}\right)$. On the other hand, we prove that every
set $\mathcal{S}$ admits a Riesz sequence $E\left(\Lambda\right)$ such that
$\Lambda$ does contain arbitrary long arithmetic progressions of length $N$ and
step $\ell=O\left(N\right)$.
