On some lacunary power series
type: Item
instance of: scholarly article
title: On some lacunary power series
zbMATH Open document ID: 1113.30004
author: Krzysztof Barański
published in: Michigan Mathematical Journal
publication date: 2007-04-05T00:00:00Z
review text: The author continues the investigations of \textit{R. Salem} and \textit{A. Zygmund} [Duke Math. J. 12, 569--578 (1945; Zbl 0060.20402), \textit{A. S. Belov} [Mat. Sb. 181, No.~8, 1048--1060 (1990; Zbl 0723.30037)], \textit{A. Canton, A. Granados}, and \textit{Ch. Pommerenke} [Mich. Math. J. 52, No.~2, 279--287 (2004; Zbl 1067.30071)]. He proves the following. Theorem. There exists \(q \in (0, 1)\) with the following property. Let \(b = 2, 3, \dots\) and let \[ f (z) = \sum \limits_{n=0}^{\infty} a_n z^{b^n}, \] where \(a_n \in \mathbb C,\) such that \(\sum \limits_{n=0}^{\infty} | a_n| < \infty\) and \(| a_{n+1}| > q | a_n| \) for sufficiently large \(n.\) Then there exists a disk \(D \subset \mathbb C\) such that \(f\) assumes every value from \(D\) uncountable many times on the unit circle. Hence, \(f\) does not preserve Borel set on the unit circle.
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 5140511
zbMATH Keywords: Peano curve
zbMATH Keywords: Cantor set
zbMATH Keywords: Borel set
zbMATH Keywords: Lusin-Curves theorem
zbMATH Keywords: Weierstrass function
zbMATH Keywords: triangle
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1144437438
OpenAlex ID: W2074698906
cites work: Q4458792
cites work: Q3211601
cites work: Borel images and analytic functions
cites work: On lacunary power series
cites work: Lacunary power series and Peano curves
DOI: 10.1307/MMJ/1144437438
description: scientific article; zbMATH DE number 5140511