Coefficients of squares of Newman polynomials

2008/06/11 by Kolountzakis, Mihail N.
#11B34 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0806.1809

Abstract

We show that there are polynomials p_N of arbitrarily large degree N, with coefficients equal to 0 or 1 (Newman polynomials), such that \liminf_{N → ∞} N \Linfp_N² / p_N²(1) lt; 1, where \Linfq denotes the maximum coefficient of the polynomial q and which, at the same time, are sparse: p_N(1)/N → 0. This disproves a conjecture of Yu \citeyu. We build on some previous results of Berenhaut and Saidak \citeberenhaut-saidak and Dubickas \citedubickas whose examples lacked the sparsity. This sparsity we create from these examples by randomization.

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