A quantitative CLT for the zeros of random trigonometric polynomials

Let $X_N(t)$, $t\in [0,\pi ]$, be a stationary Gaussian trigonometric polynomial, and let ${\mathcal{Z}}_N$ denote the number of zeros of the rescaled process $X_N(t\mathrm{/}N)$ on $[0,N\pi ]$. We establish the quantitative central limit theorem $$d_{\mathrm{W}}(\frac{{\mathcal{Z}}_N-\mathbb{E}[{\mathcal{Z}}_N]}{\sqrt{\mathrm{Var}({\mathcal{Z}}_N)}},Z)\le C\frac{(\log N{)}^2}{\sqrt{N}},$$ where $Z$ is standard Gaussian and $d_{\mathrm{W}}$ denotes the Wasserstein-1 distance. The proof combines Ornstein–Uhlenbeck regularization of the singular Kac representation with direct estimates of the covariance between the Malliavin–Stein factor and Stein test functions. A lattice lower bound of order $N^{-1\mathrm{/}2}$ shows that the convergence rate is optimal up to logarithmic factors.