Geodesic
Periodicity and non-periodicity of finite sequences
Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 275-278.

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Here we study a problem, concerned with generating pseudo-random sequences. The non-periodicity is one of crucial properties of a good pseudo-random sequence. But an infinite non-periodic sequence may have initial segment with improper behavior. For example, the decimal expansion of the Liouvillean number $$ \sum\limits_{n=0}^\infty 10^{-n!} $$ contains only a few digits, equal to one, and all the other are equal to zero. For practical purposes, therefore, we need to introduce notions of periodicity and sufficient non-periodicity of finite sequences. The paper treats certain decimal expansions of real numbers and the links between their arithmetic properties and sufficient non-periodicity of these expansions. Several ways to generate numbers with sufficiently non-periodic expansions are discussed. We overview certain results in this direction and possible ways to develop them further. We briefly describe problems with polyadic expansions. They are rather convenient since they don't involve division. The known results are decribed and certain problems formulated. Bibliography: 11 titles.
Keywords: finite periodicity, arithmetic properties. 
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V. G. Chirskii. Periodicity and non-periodicity of finite sequences. Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 275-278. http://geodesic.mathdoc.fr/item/CHEB_2017_18_2_a18/

[1] Chirskii V. G., Nesterenko A. Yu., “An approach to the transformation of periodic sequences”, Discrete Mathematics and Applications, 27:1 (2015), 1–6 | DOI | DOI | MR

[2] Chirskii V. G., “Arithmetic properties of polyadic series with periodic coefficients”, Doklady Mathematics, 90:3 (2014), 766–768 | DOI | DOI | MR | Zbl

[3] Chirskii V. G., “On the arithmetic properties of generalized hypergeometric series with irrational parameters”, Izvestiya Mathematics, 2014, no. 6, 1244–1260 | DOI | DOI | MR | Zbl

[4] Chirskii V. G., “Arithmetic properties of polyadic series with periodic coefficients”, Izvestiya Mathematics, 81:2 (2017), 444–461 | DOI | DOI | MR | Zbl

[5] Chirskii V. G., “An approach to the transformation of periodic sequences”, Chebushevskii sb., 17:3 (2016), 180–185

[6] Chirskii V. G., “Arithmetic properties of polyadic numbers”, Chebushevskii sb., 16:1 (2015), 254–264

[7] Bertrand D., Chirskii V. G., Yebbou Y., “Effective estimates for global relations on Euler-type series”, Ann. Fac. Sci. Toulouse, XIII:2 (2004), 241–260 | DOI | MR | Zbl

[8] Chirskii V. G., “Arithmetic properties of Euler series”, Moscow University Mathematics Bulletein, 70:1 (2015), 41–43 | DOI | MR | Zbl

[9] Chirskii V. G., Matveev V. Yu., “On some properties of polyadic expansions”, Chebushevskii sb., 14:2 (2013), 164–172

[10] Chirskii V. G., Matveev V. Yu., “On the representation of natural numbers”, Chebushevskii sb., 14:1 (2013), 92–101

[11] Chirskii V. G., Matveev V. Yu., “On the representation of natural numbers”, MSU Bulletin, 2013, no. 6, 57–59