Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas¹

¹ Department of Mathematics and Informatics Vilnius University Naugarduko 24 Vilnius LT-03225, Lithuania and Vilnius University Institute of Mathematics and Informatics Vilnius University Akademijos 4 Vilnius LT-08663, Lithuania

Annales Polonici Mathematici, Tome 105 (2012) no. 2, pp. 145-153.

Résumé

Let $P$ be a unimodular polynomial of degree $d-1$. Then the height $H(P^2)$ of its square is at least $\sqrt{d/2}$ and the product $L(P^2)H(P^2)$, where $L$ denotes the length of a polynomial, is at least $d^2$. We show that for any $\varepsilon>0$ and any $d \geq d(\varepsilon)$ there exists a polynomial $P$ with $\pm 1$ coefficients of degree $d-1$ such that $H(P^2)(2+\varepsilon)\sqrt{d \log d}$ and $L(P^2)H(P^2)(16/3+\varepsilon) d^2 \log d$. A similar result is obtained for the series with $\pm 1$ coefficients. Let $A_m$ be the $m$th coefficient of the square $f(x)^2$ of a unimodular series $f(x)=\sum_{i=0}^{\infty} a_i x^i$, where all $a_i \in \mathbb C$ satisfy $|a_i|=1$. We show that then $\limsup_{m \to \infty} |A_m|/\sqrt{m} \geq 1$ and that there exist some infinite series with $\pm 1$ coefficients and an integer $m(\varepsilon)$ such that $|A_m| (2+\varepsilon)\sqrt{m \log m}$ for each $m \geq m(\varepsilon)$.

Détail

DOI : 10.4064/ap105-2-3

Keywords: unimodular polynomial degree d height its square least sqrt product where denotes length polynomial least varepsilon geq varepsilon there exists polynomial coefficients degree d varepsilon sqrt log varepsilon log similar result obtained series coefficients mth coefficient square unimodular series sum infty i where mathbb satisfy limsup infty sqrt geq there exist infinite series coefficients integer varepsilon varepsilon sqrt log each geq varepsilon