  Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different
properties of dilations systems of a fixed function $\varphi \in L^2(0,1)$.
More precisely, we study when the system $\{\varphi(nx)\}_n$ is a Bessel
sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able
to characterize these properties in terms of multipliers of the Hardy space
$\mathcal{H}^2$ of Dirichtet series and, also, in terms of Hardy spaces on the
infinite polytorus. We also address the multivariate case.
