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

Abstract: 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