On the discrepancy of Markov-normal sequences
Journal de théorie des nombres de Bordeaux, Volume 8 (1996) no. 2, p. 413-428

We construct a Markov normal sequence with a discrepancy of O(N -1/2 log 2 N). The estimation of the discrepancy was previously known to be O(e -c(logN) 1/2 ).

On construit une suite normale de Markov dont la discrépance est O(N -1/2 log 2 N), améliorant en cela un résultat donnant l’estimation O(e -c(logN) 1/2 ).

@article{JTNB_1996__8_2_413_0,
     author = {Levin, Mordechay B.},
     title = {On the discrepancy of Markov-normal sequences},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {8},
     number = {2},
     year = {1996},
     pages = {413-428},
     zbl = {0916.11044},
     mrnumber = {1438479},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1996__8_2_413_0}
}
Levin, M. B. On the discrepancy of Markov-normal sequences. Journal de théorie des nombres de Bordeaux, Volume 8 (1996) no. 2, pp. 413-428. http://www.numdam.org/item/JTNB_1996__8_2_413_0/

[1] A.G. Postnikov and I.I. Piatetski-Shapiro, A Markov sequence of symbols and a normal continued fraction, Izv. Akad. Nauk SSSR, Ser. Mat., 1957, v.21, 501-514. | MR 101856 | Zbl 0078.31102

[2] M. Smorodinsky and B. Weiss, Normal sequences for Markov shifts and intrinsically ergodic subshifts, Israel Journal of Mathematics, 1987, v.59, 225-233. | MR 920084 | Zbl 0643.10041

[3] A. Bertrand-Mathis, Points generiques de Champernowne sur certains systems codes; application aux θ-shifts, Ergod. Th. & Dynam. Sys.,1988, v. 8, 35-51. | Zbl 0657.28014

[4] N.N. Chentsov, Pseudorandom numbers for modeling Markov chains, U.S.S.R. Comput. Maths. Math. Phis., 1967, vol. 7, no 3, 218-233. | Zbl 0181.45105

[5] N.M. Korobov, Exponential sums and their applications, Dordrecht, 1992, 209 pages. | MR 1162539 | Zbl 0754.11022

[6] U.N. Sahov, Imitation of simplist Markov processes, Izv. Akad. Nauk SSSR, Ser. Mathem., 1959, v.23, 815-822. | MR 114241

[7] U.N. Sahov, The construction of sequence of signs that is normal in the sen se of Markov, Moskovskii Gosudarstvennyi Pedagogiceskii institute im. V.I. Lenina, Ucenye Zapiski, 1971, v. 375,143-155.

[8] W. Feller, An Introduction to Probability Theory and Its Applications, vol.1, New York, 1965. | Zbl 0039.13201

[9] J.L. Kemeny and J.L. Snell, Finite Markov chains, New York, 1960, 210 pages. | MR 115196 | Zbl 0089.13704

[10] I.S. Berezin and N.P. Zhidkov, Computing methods, vol. 2, Pergamon Press, Oxford, 1965, 267, 268. | Zbl 0122.12903

[11] N.M. Korobov, Distribution of fractional parts of exponential function, Vestnic Moskov. Univ.,Ser.1 Mat. Meh., 1966, v. 21, no. 4, 42-46. | MR 197435 | Zbl 0154.04801

[12] M.B. Levin, The distribution of fractional parts of the exponential function, Soviet. Math. (Iz. Vuz.), 1977, v. 21, no.11, 41-47. | MR 506058 | Zbl 0389.10037

[13] U.N. Sahov, Some bounds in the construction of Bernoulli-normal sequences of signs, Math. Notes, 1971, v. 10, 724-730. | Zbl 0248.65071

[14] M.B. Levin, On normal sequence for Markov and Bernoulli shifts, 49-53, Proccedings of the Israel Mathematical Union Conference,1994, Beer Sheva, 97-100.

[15] W. Philipp, Limit theorems for lacunary series and uniform distribution mod 1, Acta Arithm., 1975, v. 26, 241-251. | MR 379420 | Zbl 0263.10020