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

2024/07/22 by Bedert, Benjamin · 1 citation
#11C08 #26C10 #30C15 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2407.16075

Abstract

Let Z(N) denote the minimum number of zeros in [0,2π] that a cosine polynomial of the form f_A(t)=∑_{n∈ 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 (loglog N)^(1+o(1)) which exponentially improves on the previous best bounds of the form Z(N)\geqslant (logloglog N)^c due to Erdélyi and Sahasrabudhe.

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