Geodesic
On the Behavior of the Spectrum of the Limit Frequencies of Digits under Perturbations of a Real Number
Matematičeskie zametki, Tome 86 (2009) no. 6, pp. 884-891 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown that, under the action of random digitwise decaying perturbations not changing the spectrum of the limit frequencies of $r$-adic digits, an arbitrary real number written in the $r$-adic number system becomes almost surely normal over all bases multiplicatively independent of $r$.
Keywords: $r$-adic number system, normal number, digitwise perturbations of a number, spectrum of limit frequencies of digits
Mots-clés : Lebesgue measure. 
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L. N. Pushkin. On the Behavior of the Spectrum of the Limit Frequencies of Digits under Perturbations of a Real Number. Matematičeskie zametki, Tome 86 (2009) no. 6, pp. 884-891. http://geodesic.mathdoc.fr/item/MZM_2009_86_6_a6/

[1] M. E. Borel, “Les probabilités dénombrables et leurs applications arithmétiques”, Rend. Circ. Mat. Palermo, 27:1 (1909), 247–271 | DOI | Zbl

[2] J. W. S. Cassels, “On a problem of Steinhaus about normal numbers”, Colloq. Math., 7 (1959), 95–101 | MR | Zbl

[3] W. M. Schmidt, “On normal numbers”, Pacific J. Math., 10 (1960), 661–672 | MR | Zbl

[4] C. M. Colebrook, J. H. B. Kemperman, “On non-normal numbers”, Indag. Math., 30 (1968), 1–11 | MR | Zbl

[5] B. Volkmann, “On the Cassels–Schmidt theorem. I”, Bull. Sci. Math. (2), 108:3 (1984), 321–336 | MR | Zbl

[6] L. N. Pushkin, “Metricheskii variant teoremy Kasselsa–Shmidta”, Matem. zametki, 46:1 (1989), 60–66 | MR | Zbl

[7] J. Feldman, M. Smorodinsky, “Normal numbers from independent processes”, Ergodic Theory Dynam. Systems, 12:4 (1992), 707–712 | MR | Zbl

[8] L. Keipers, G. Niderreiter, Ravnomernoe raspredelenie posledovatelnostei, Nauka, M., 1985 | MR | Zbl

[9] I. I. Shapiro-Pyatetskii, “O zakonakh raspredeleniya drobnykh dolei pokazatelnoi funktsii”, Izv. AN SSSR. Ser. matem., 15:1 (1951), 47–52 | MR | Zbl

[10] E. Lukach, Kharakteristicheskie funktsii, Nauka, M., 1979 | MR | Zbl

[11] H. Davenport, P. Erdős, W. J. LeVeque, “On Weyl's criterion for uniform distribution”, Michigan Math. J., 10:3 (1963), 311–314 | DOI | MR | Zbl

[12] A. Baker, Transcendental Number Theory, Cambridge Univ. Press, London, 1975 | MR | Zbl