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

TL;DR

This paper investigates extremal properties of cosine sums and Newman polynomials, providing exact values for small cases and bounds for larger ones, highlighting the computational challenges involved.

Contribution

It determines exact values of (2) and (3), and offers bounds for (5), advancing understanding of finite search problems in cosine sums and Newman polynomials.

Findings

Exact values for (2) and (3)

Bounds on (5)

Discussion on computational complexity

Abstract

A length $n$ cosine sum is an expression of the form $\cos a_1\theta + \cdots + \cos a_n\theta$ 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 $-\lambda(n)$ 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 $\mu(n)$ 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 $\lambda(n)$ or $\mu(n)$ for a given $n$ in finitely many steps. Campbell et al. found the value of $\mu(3)$ in 1983, and Goddard found the value of $\mu(4)$ in 1992. In this paper, we find the values of $\lambda(2)$ and $\lambda(3)$ and nontrivial bounds on $\mu(5)$. We also include further remarks on the seemingly difficult general task of reducing the computation of $\lambda(n)$ or $\mu(n)$ to a finite problem.