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Hausdorff dimension of the set of almost convergent sequences

Part of: Special methods of summability Normed linear spaces and Banach spaces; Banach lattices Probabilistic theory: distribution modulo $1$; metric theory of algorithms Special classes of linear operators

Published online by Cambridge University Press:  06 January 2022

Alexandr Usachev
Alexandr Usachev*
Affiliation:
School of Mathematics and Statistics, Central South University, Hunan, 410085, China and Faculty of Mathematics, Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia. E-mail: alex.usachev.ru@gmail.com
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Abstract

The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

Keywords

Hausdorff dimensionbinary expansionalmost convergence

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension 40G99: None of the above, but in this section
Secondary: 46B45: Banach sequence spaces 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 691 - 697
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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