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On expressible sets and p-adic numbers

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  25 February 2011

Jaroslav Hančl ,
Radhakrishnan Nair ,
Simona Pulcerova  and
Jan Šustek
Jaroslav Hančl
Affiliation:
Department of Mathematics and Institute for Research and Applications of Fuzzy Modelling, University of Ostrava, 30 dubna 22, 701 03 Ostrava 1, Czech Republic, (hancl@osu.cz; jan.sustek@seznam.cz)
Radhakrishnan Nair
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK (nair@liverpool.ac.uk)
Simona Pulcerova
Affiliation:
Department of Mathematical Methods in Economics, Faculty of Economics, VŠB—Technical University of Ostrava, Sokolská třída 33, 701 21 Ostrava 1, Czech Republic (simona.sobkova@vsb.cz)
Jan Šustek
Affiliation:
Department of Mathematics and Institute for Research and Applications of Fuzzy Modelling, University of Ostrava, 30 dubna 22, 701 03 Ostrava 1, Czech Republic, (hancl@osu.cz; jan.sustek@seznam.cz)
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Abstract

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Continuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.

Keywords

expressible setp-adic numbersKhinchin–Lutz Theorem

MSC classification

Secondary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 411 - 422
Copyright
Copyright © Edinburgh Mathematical Society 2011

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