Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas

Annales Polonici Mathematici, Tome 105 (2012), p. 145-153 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $f (x) = \sum_{i = 0}^{\infty} a_{i} x^{i}$ , where all $a_{i} \in ℂ$ satisfy $| a_{i} | = 1$ . We show that then $l i m s u p_{m \to \infty} | A_{m} | / \sqrt m \geq 1$ and that there exist some infinite series with ±1 coefficients and an integer m(ε) such that $| A_{m} | < (2 + ε) \sqrt (m l o g m)$ for each m ≥ m(ε).

Publié le : 2012-01-01

Zbl 1277.11099

EUDML-ID : urn:eudml:doc:280396

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Artūras Dubickas. Heights of squares of Littlewood polynomials and infinite series. Annales Polonici Mathematici, Tome 105 (2012) pp. 145-153. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap105-2-3/