Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

On expressible sets and p-adic numbers

Jaroslav Hančla1, Radhakrishnan Naira2, Simona Pulcerovaa3 and Jan Šusteka1

a1 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)

a2 Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK (nair@liverpool.ac.uk)

a3 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)

Abstract

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.

(Received January 14 2009)

(Online publication February 25 2011)

Keywords

  • expressible set;
  • p-adic numbers;
  • Khinchin–Lutz Theorem

2010 Mathematics subject classification

  • Primary 11K55