Consider the Hilbert space $H=L^2(-\pi,\pi)$. A subset of $H$ is said to be total in $H$ if the closure of its span is the whole $H$.
For instance, the Fourier basis $\{e^{in x}\}_{n\in\mathbb{Z}}$ is a Hilbert basis for $H$ and in particular it is total in $H$.
I am interested in families of the form $\{e^{is_n x}\}_{n\in\mathbb{Z}}$, where $\mathcal{S}=\{s_n\}_{n\in\mathbb{Z}}$ is a strictly increasing sequence. Is there any easy condition on $\mathcal{S}$ which guarantees that such a family is total in $H$? I would guess that $S$ should at least be diverging, in order to capture arbitrarily high frequencies, but I'm not sure that it is actually a necessary condition.
Of course not every increasing sequence $\mathcal{S}$ has this property. An easy counter example is any $\mathcal{S}\subsetneq \mathbb{Z}$, since the obtained family would be a proper subset of the Fourier basis. However I guess there must be some easy and quite general criterion to check whether a sequence gives rise to a family total in $H$.
This is a classical and widely studied question. See the answers to https://mathoverflow.net/questions/224001/completeness-of-nonharmonic-fourier-series.
Roughly speaking, you get totality if $s_n$ grows slower than the Fourier basis (i.e. less than linearly with rate $1/(2\pi)$)