Typical Sequence of Real Numbers From the Unit Interval Has All Distribution Functions

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József Bukor, Kálmán Liptai, János T. Tóth


This note is devoted to the study of typical properties (in Baire category sense) of sequences of real numbers in [0, 1]. We prove that the subset of sequences that have all distribution functions forms a residual set.

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  1. J. Bukor, J. T. Tóth, On Topological Properties of the Set of Maldistributed Sequences, Acta Univ. Sapient. Math. 12 (2020), 272–279. https://doi.org/10.2478/ausm-2020-0018.
  2. P. Kostyrko, M. Macaj, T. Šalát, O. Strauch, On Statistical Limit Points, Proc. Amer. Math. Soc. 129 (2000), 2647–2654. https://doi.org/10.1090/s0002-9939-00-05891-3.
  3. V. László, T. Šalát, The Structure of Some Sequence Spaces, and Uniform Distribution (Mod 1), Period Math. Hung. 10 (1979), 89–98. https://doi.org/10.1007/bf02018376.
  4. L. Olsen, Extremely Non-Normal Numbers, Math. Proc. Camb. Phil. Soc. 137 (2004), 43–53. https://doi.org/10.1017/s0305004104007601.
  5. J. Hyde, V. Laschos, L. Olsen, T. Petrykiewicz, A. Shaw, Iterated Cesàro Averages, Frequencies of Digits, and Baire Category, Acta Arith. 144 (2010), 287–293. https://doi.org/10.4064/aa144-3-6.
  6. J.C. Oxtoby, Measure and Category, Springer, New York, 1996.
  7. T. Šalát, A Remark on Normal Numbers, Rev. Roumaine Math. Pures Appl. 11 (1966), 53–56.
  8. T. Šalát, On Statistically Convergent Sequences of Real Numbers, Math. Slovaca, 30 (1980), 139–150. http://dml.cz/dmlcz/136236.
  9. O. Strauch, Uniformly Maldistributed Sequences in a Strict Sense, Monatsh. Math. 120 (1995), 153–164. https://doi.org/10.1007/bf01585915.
  10. O. Strauch, Distribution of Sequences: A Theory, VEDA, Bratislava; Academia, Prague, 2019.
  11. O. Strauch, Š. Porubský, Distribution of Sequences: A Sampler, Peter Lang, Frankfurt am Main, 2005.