Heights of squares of Littlewood polynomials and infinite series
Volume 105 / 2012
Abstract
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 $m \geq m(\varepsilon)$.
