On the discrepancy of Markov-normal sequences
Journal de Théorie des Nombres de Bordeaux, Volume 8 (1996) no. 2, pp. 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 = {M. B. Levin},
     title = {On the discrepancy of {Markov-normal} sequences},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {413--428},
     publisher = {Universit\'e Bordeaux I},
     volume = {8},
     number = {2},
     year = {1996},
     doi = {10.5802/jtnb.178},
     zbl = {0916.11044},
     mrnumber = {1438479},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.178/}
}
TY  - JOUR
TI  - On the discrepancy of Markov-normal sequences
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 1996
DA  - 1996///
SP  - 413
EP  - 428
VL  - 8
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.178/
UR  - https://zbmath.org/?q=an%3A0916.11044
UR  - https://www.ams.org/mathscinet-getitem?mr=1438479
UR  - https://doi.org/10.5802/jtnb.178
DO  - 10.5802/jtnb.178
LA  - en
ID  - JTNB_1996__8_2_413_0
ER  - 
%0 Journal Article
%T On the discrepancy of Markov-normal sequences
%J Journal de Théorie des Nombres de Bordeaux
%D 1996
%P 413-428
%V 8
%N 2
%I Université Bordeaux I
%U https://doi.org/10.5802/jtnb.178
%R 10.5802/jtnb.178
%G en
%F JTNB_1996__8_2_413_0
M. B. Levin. On the discrepancy of Markov-normal sequences. Journal de Théorie des Nombres de Bordeaux, Volume 8 (1996) no. 2, pp. 413-428. doi : 10.5802/jtnb.178. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.178/

[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

Cited by Sources: