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

## TL;DR

This paper investigates the expected number of real zeros of random cosine polynomials with dependent coefficients arranged in pairwise equal blocks, revealing that certain block structures increase the expected zeros beyond classical results.

## Contribution

It introduces new classes of random cosine polynomials with pairwise equal blocks of coefficients and analyzes their expected real zeros, showing increased zeros in specific cases.

## Key findings

- Polynomials with fixed-length equal blocks have the same expected zeros as classical models.
- Two-block coefficient structures lead to significantly more expected real zeros.

## Abstract

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.