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On expressible sets and p-adic numbers

  • Jaroslav Hančl (a1), Radhakrishnan Nair (a2), Simona Pulcerova (a3) and Jan Šustek (a1)

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.

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1.Apéry R., Irrationalité de ζ(2) et ζ(3), Astérisque 61 (1979), 1113.
2.Asmar N. H. and Nair R., Certain averages on the a-adic numbers, Proc. Am. Math. Soc. 114(1) (1992), 2128.
3.Beresnevich V., Dickinson H. and Velani S., Measure theoretic laws for lim sup sets, Memoirs of the American Mathematical Society, Volume 179 (American Mathematical Society, Providence, RI, 2006).
4.Bernik V. I. and Dodson M. M., Metric Diophantine approximation on manifolds, Cambridge Tracts in Mathematics, Volume 137 (Cambridge University Press, 1999).
5.Bodiagin D., Hančl J., Nair R. and Rucki P., On summing to arbitrary real numbers, Elem. Math. 63(1) (2008), 3034.
6.Cassels J. W. S., Local fields, London Mathematical Society Student Texts, Volume 3 (Cambridge University Press, 1986).
7.Erdős P., Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975), 17.
8.Hančl J., Expression of real numbers with the help of infinite series, Acta Arith. 59 (1991), 97104.
9.Hančl J., Two criteria for transcendental sequences, Matematiche (Catania) 56(1) (2001), 149160.
10.Hančcl J., Liouville sequences, Nagoya Math. J. 172 (2003), 173187.
11.Hančl J. and Filip F., Irrationality measure of sequences, Hiroshima Math. J. 35(2) (2005), 183195.
12.Hančl J., Nair R. and Šustek J., On the Lebesgue measure of the expressible set of certain sequences, Indagationes Math. 17(4) (2006), 567581.
13.Hančl J. and Šustek J., Expressible sets of certain sequences with Hausdorff dimension zero, Monatsh. Math. 152(4) (2007), 315319.
14.Lutz É., Sur les approximations diophantiennes linéaires P-adiques, Actualités Scientifiques Et Industrielles, Volume 1224 (Hermann, Paris, 1955).
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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