Finite searches, Chowla's cosine problem, and large Newman polynomials

Abstract

A length $n$ cosine sum is an expression of the form $\cos a_1θ+ \cdots + \cos a_nθ$ where $a_1 < \cdots < a_n$ are positive integers, and a length $n$ Newman polynomial is an expression of the form $z^{a_1} + \cdots + z^{a_n}$ where $a_1 < \cdots < a_n$ are nonnegative integers. We define $-λ(n)$ to be the largest minimum of a length $n$ cosine sum as $\{a_1,\ldots,a_n\}$ ranges over all sets of $n$ positive integers, and we define $μ(n)$ to be the largest minimum modulus on the unit circle of a length $n$ Newman polynomial as $\{a_1,\ldots,a_n\}$ ranges over all sets of $n$ nonnegative integers. Since there are infinitely many possibilities for the $a_j$, it is not obvious how to compute $λ(n)$ or $μ(n)$ for a given $n$ in finitely many steps. Campbell et al. found the value of $μ(3)$ in 1983, and Goddard found the value of $μ(4)$ in 1992. In this paper, we find the values of $λ(2)$ and $λ(3)$ and nontrivial bounds on $μ(5)$. We also include further remarks on the seemingly difficult general task of reducing the computation of $λ(n)$ or $μ(n)$ to a finite problem.