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Hausdorff dimension of the set of almost convergent sequences

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

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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References

Albeverio, S., Pratsiovytyi, M. and Torbin, G., Topological and fractal properties of real numbers which are not normal, Bull. Sci. Math. 129 (2005), 615–630.CrossRefGoogle Scholar
Besicovitch, A. S., On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1935), 321–330.CrossRefGoogle Scholar
Boos, J., Classical and modern methods in summability, Oxford Mathematical Monographs (Oxford University Press, Oxford, 2000).Google Scholar
Cardone, G., Corbo Esposito, A. and Faella, L., Hausdorff dimension for level sets of upper and lower limits of generalized averages of binary digits, Math. Methods Appl. Sci. 29 (2006), 1983–2008.CrossRefGoogle Scholar
Chen, H., Wang, X. and Wen, Z., Multifractal analysis on the level sets described by moving averages, Int. J. Number Theory 11 (2015), 2175–2189.CrossRefGoogle Scholar
Connor, J., Almost none of the sequences of 0’s and 1’s are almost convergent, Internat. J. Math. Math. Sci. 13 (1990), 775–777.Google Scholar
Eggleston, H. G., The fractional dimension of a set defined by decimal properties, Quart. J. Math., Oxford Ser. 20 (1949), 31–36.CrossRefGoogle Scholar
Falconer, K., Fractal geometry, 2nd edition (John Wiley & Sons, Inc., Hoboken, NJ, 2003). Mathematical foundations and applications.CrossRefGoogle Scholar
Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167–190.CrossRefGoogle Scholar
Semenov, E. and Sukochev, F., Extreme points of the set of Banach limits, Positivity 17 (2013), 163–170.Google Scholar
Semenov, E., Sukochev, F. and Usachev, A., Geometry of Banach limits and their applications, Russian Math. Surveys 75(4) (2020), 725–763.CrossRefGoogle Scholar
Sofi, M. A., Banach limits: Some new thoughts and perspectives, J. Anal. 29 (2021), 591–606 CrossRefGoogle Scholar
Sukochev, F., Usachev, A. and Zanin, D., Dixmier traces generated by exponentiation invariant generalised limits, J. Noncommut. Geom. 8(2) (2014), 321–336.CrossRefGoogle Scholar
Sukochev, F., Usachev, A. and Zanin, D., Generalized limits with additional invariance properties and their applications to noncommutative geometry, Adv. Math. 239 (2013), 164–189.CrossRefGoogle Scholar