A quantitative CLT for the zeros of random trigonometric polynomials

Let $X_N(t)$, $t\in[0,π]$, be a stationary Gaussian trigonometric polynomial, and let $\mathcal Z_N$ denote the number of zeros of the rescaled process $X_N(t/N)$ on $[0,Nπ]$. We establish the quantitative central limit theorem \[ d_{\mathrm W}\left( \frac{\mathcal Z_N-\mathbb E[\mathcal Z_N]}