Real zeros of random trigonometric polynomials with $ \ell $-periodic coefficients

The large degree asymptotics of the expected number of real zeros of a random trigonometric polynomial $$T_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) + b_j \sin (j x), \ x \in (0,2\pi),$$ with i.i.d. real-valued standard Gaussian coefficients is known to be $2n / \sqrt{3}$. In this article, we consider quite a different and extreme setting on the set of the coefficients of $T_n$. We show that a random trigonometric polynomial of degree $n$ with $\ell$-periodic i.i.d. Gaussian coefficients is expected to have significantly more real zeros compared to the classical case with i.i.d. Gaussian coefficients. More precisely, the expected number of real zeros of $T_n$ is proportional to $n$ with a proportionality constant $\mathrm{C}_{\ell,r} \in (\sqrt{2},2]$, which is explicitly represented by a double integral formula. The case $r=0$ is marked as a special one since in such a case $T_n$ asymptotically obtains the largest possible number of real zeros