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• # Artykuł - szczegóły

## Annales Polonici Mathematici

2012 | 105 | 2 | 145-153

## Heights of squares of Littlewood polynomials and infinite series

EN

### Abstrakty

EN
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).

145-153

wydano
2012

### Twórcy

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