Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas. "Heights of squares of Littlewood polynomials and infinite series." Annales Polonici Mathematici 105.2 (2012): 145-153. <http://eudml.org/doc/280396>.

@article{ArtūrasDubickas2012,
abstract = {Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_m$ be the mth coefficient of the square f(x)² of a unimodular series $f(x) = ∑_\{i=0\}^\{∞\} a_i x^i$, where all $a_i ∈ ℂ$ satisfy $|a_i| = 1$. We show that then $lim sup_\{m → ∞\} |A_m|/√m ≥ 1$ and that there exist some infinite series with ±1 coefficients and an integer m(ε) such that $|A_m| < (2+ε)√(mlogm)$ for each m ≥ m(ε).},
author = {Artūras Dubickas},
journal = {Annales Polonici Mathematici},
keywords = {Littlewood polynomial; height; length; Hoeffding's inequality},
language = {eng},
number = {2},
pages = {145-153},
title = {Heights of squares of Littlewood polynomials and infinite series},
url = {http://eudml.org/doc/280396},
volume = {105},
year = {2012},
}

TY - JOUR
AU - Artūras Dubickas
TI - Heights of squares of Littlewood polynomials and infinite series
JO - Annales Polonici Mathematici
PY - 2012
VL - 105
IS - 2
SP - 145
EP - 153
AB - Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_m$ be the mth coefficient of the square f(x)² of a unimodular series $f(x) = ∑_{i=0}^{∞} a_i x^i$, where all $a_i ∈ ℂ$ satisfy $|a_i| = 1$. We show that then $lim sup_{m → ∞} |A_m|/√m ≥ 1$ and that there exist some infinite series with ±1 coefficients and an integer m(ε) such that $|A_m| < (2+ε)√(mlogm)$ for each m ≥ m(ε).
LA - eng
KW - Littlewood polynomial; height; length; Hoeffding's inequality
UR - http://eudml.org/doc/280396
ER -