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Let $X_N$ be a random trigonometric polynomial of degree $N$ with iidcoefficients and let $Z_N(I)$ denote the (random) number of its zeros lying inthe compact interval $I\subset\mathbb{R}$. Recently, a number of importantadvances were made in the understanding of the asymptotic behaviour of $Z_N(I)$as $N\to\infty$, in the case of standard Gaussian coefficients. The maintheorem of the present paper is a universality result, that states that thelimit of $Z_N(I)$ does not really depend on the exact distribution of thecoefficients of $X_N$. More precisely, assuming that these latter are iid withmean zero and unit variance and have a density satisfying certain conditions,we show that $Z_N(I)$ converges in distribution toward $Z(I)$, the number ofzeros within $I$ of the centered stationary Gaussian process admitting thecardinal sine for covariance function.

Local universality of the number of zeros of random trigonometric polynomials with continuous coefficients