Real zeros of random trigonometric polynomials with pairwise equal blocks of coefficients

It is well known that the expected number of real zeros of a random cosine polynomial $V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2\pi)$, with the $a_j$ being standard Gaussian i.i.d. random variables is asymptotically $2n / \sqrt{3}$. On the other hand, some of the previous works on the random cosine polynomials with dependent coefficients show that such polynomials have at least $2n / \sqrt{3}$ expected real zeros lying in one period. In this paper we investigate two classes of random cosine polynomials with pairwise equal blocks of coefficients. First, we prove that a random cosine polynomial with the blocks of coefficients being of a fixed length and satisfying $A_{2j}=A_{2j+1}$ possesses the same expected real zeros as the classical case. Afterwards, we study a case containing only two equal blocks of coefficients, and show that in this case significantly more real zeros should be expected compared to those of the classical case.