An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials

TL;DR

This work addresses Littlewood's problem on the minimum number of zeros of cosine polynomials $f_A(t)=\sum_{n\in A}\cos nt$ with a fixed cardinality $|A|=N$. The authors develop an improved lower bound by fusing an exponential $L^1$-type bound for structured trigonometric polynomials with a refined structural decomposition that yields highly periodic coefficient blocks, alongside the Littlewood $L^1$ conjecture. The key technical advances are an $L^1$ bound linking sign changes, coefficient-structure, and the zero count, and a sharpened structural result that reduces the period from previous double-exponential bounds to $P=\exp(O(d\log\log d))$, enabling a near-optimal growth rate in $\log\log N$. Consequently, they prove $Z(N) \ge (\log\log N)^{1-o(1)}$, significantly narrowing the gap between known lower and upper bounds and advancing the understanding of zero distribution in cosine polynomials.

Abstract

Let $Z(N)$ denote the minimum number of zeros in $[0,2π]$ that a cosine polynomial of the form $$f_A(t)=\sum_{n\in A}\cos nt$$ can have when $A$ is a finite set of non-negative integers of size $|A|=N$. It is an old problem of Littlewood to determine $Z(N)$. In this paper, we obtain the lower bound $Z(N)\geqslant (\log\log N)^{(1+o(1))}$ which exponentially improves on the previous best bounds of the form $Z(N)\geqslant (\log\log\log N)^c$ due to Erdélyi and Sahasrabudhe.

Theorems & Definitions (20)

• Theorem 1.2: erdelyi, Theorem 2.1
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Littlewood's $L^1$ conjecture
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4