  In this paper, we investigate the local universality of the number of zeros
of a random periodic signal of the form $S_n(t)=\sum_{k=1}^n a_k f(k t)$, where
$f$ is a $2\pi-$periodic function satisfying weak regularity conditions and
where the coefficients $a_k$ are i.i.d. random variables, that are centered
with unit variance. In particular, our results hold for continuous piecewise
linear functions. We prove that the number of zeros of $S_n(t)$ in a shrinking
interval of size $1/n$ converges in law as $n$ goes to infinity to the number
of zeros of a Gaussian process whose explicit covariance only depends on the
function $f$ and not on the common law of the random coefficients $(a_k)$. As a
byproduct, this entails that the point measure of the zeros of $S_n(t)$
converges in law to an explicit limit on the space of locally finite point
measures on $\mathbb R$ endowed with the vague topology. The standard tools
involving the regularity or even the analyticity of $f$ to establish such kind
of universality results are here replaced by some high-dimensional Berry-Esseen
bounds recently obtained in [CCK17]. The latter allow us to prove functional
CLT's in $C^1$ topology in situations where usual criteria can not be applied
due to the lack of regularity.
