Variations on Salem--Zygmund results for random trigonometric polynomials. Application to almost sure nodal asymptotics

On a probability space $(\Omega, \mathcal F, \mathbb P)$ we consider two independent sequences $(a_k)_{k \geq 1}$ and $(b_k)_{k \geq 1}$ of i.i.d. random variables that are centered with unit variance and which admit a moment strictly higher than two. We define the associated random trigonometric polynomial \[ f_n(t) :=\frac{1}{\sqrt{n}} \sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt), \quad t \in \mathbb R. \] In their seminal work, for Rademacher coefficients, Salem and Zygmund showed that $\mathbb P$ almost surely: \[ \forall t\in\mathbb R,\quad \frac{1}{2\pi}\int_{0}^{2\pi} \exp\left(i t f_n(x)\right) dx \xrightarrow[n\to\infty]~e^{-\frac{t^2}{2}}. \] In other words, if $X$ denotes an independent random variable uniformly distributed on $[0,2\pi]$, $\mathbb{P}$ almost surely, under the law of $X$, $f_n(X)$ converges in distribution to a standard Gaussian variable. In this paper, we revisit the above result from different perspectives. Namely, i) we establish a possibly sharp convergence rate for some adequate metric via the Stein's method, ii) we prove a functional counterpart of Salem--Zygmund CLT, iii) we extend it to more general distributions for $X$, iv) we also prove that the convergence actually holds in total variation. As an application, in the case where the random coefficients have a symmetric distribution and admit a moment of order $4$, we show that $\mathbb{P}$ almost surely, for any interval $[a,b] \subset [0, 2\pi]$ \[\frac{\mathcal N(f_n,[a,b])}{n} \xrightarrow[n \to +\infty]{} \frac{(b-a)}{\pi \sqrt{3}},\] where $\mathcal N(f_n,[a,b])$ denotes the number of real zeros of $f_n$ in the interval $[a,b]$. To the best of our knowledge, such an almost sure result is new in the framework of random trigonometric polynomials, even in the case of Gaussian coefficients.