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Symbolic representation of piecewise linear functions on the unit interval and application to discrepancy. (English) Zbl 0813.11045

Let \(D_ N^*(U)\) denote the star-discrepancy of a sequence \(U\) in the unit interval. “Self-similar sequences” \(U\) are known for having a small discrepancy, i.e. \(L(U) : = \lim \sup ND_ N^*(U)/ \log N\) is finite, under some additional assumptions. In this paper the author applies techniques concerning substitutions on finite alphabets and automata to prove upper bounds for \(L(U)\), in a special case of such sequences. In an appendix (by H. Faure) the best possible values for the discrepancy of van der Corput and \((n \alpha)\)-sequences are given.
Reviewer: R.F.Tichy (Graz)

MSC:

11K38 Irregularities of distribution, discrepancy
11B85 Automata sequences
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[1] Bejian, R., Minoration de la discrépance à l’origine d’une suite quelconque, Ann. Fac. Sci. Toulouse Math., 1, 201-213 (1979) · Zbl 0426.10039
[2] Borel, J.-P., Suites ayant de bonnes discrépances, J. Arith. Valenciennes (1982)
[3] Borel, J.-P., Self-similar measures and sequences, J. Number Theory, 31, 208-241 (1989) · Zbl 0673.10039
[4] Borel, J.-P., Contribution à l’étude de la distribution des suites, (Thèse d’Etat (1989), Marseille-Luminy)
[5] Borel, J.-P., Sur une suite de fonctions liées à un processus itératif, Publ. Dép. Math. Limoges, 12, 1-16 (1990)
[7] Borel, J.-P.; Pages, G.; Xiao, Y.-J., Suites à discrépance faible et intégration numérique, (Bouleau, N.; Talay, D., Probabilités numériques (1992), INRIA: INRIA Versailles), 7-22
[8] Dupain, Y., Discrépance à l’origine de la suite \((n(1+√ 5/2))\),, Ann. Inst. Fourier, 29, 81-106 (1979) · Zbl 0386.10021
[9] Eilenberg, S., Automata, Languages and Machines, Vol. A (1974), Academic Press: Academic Press London · Zbl 0317.94045
[10] Faure, H., Discrépance de suites associées à un système de numération (en dimension un), Bull. Soc. Math. France, 109, 143-182 (1981) · Zbl 0488.10052
[11] Faure, H., Etude des restes pour les suites de van der Corput généralisées, J. Number Theory, 16, 376-394 (1983) · Zbl 0513.10047
[12] Faure, H., Good permutations for extreme discrepancy, J. Number Theory, 42, 47-56 (1992) · Zbl 0768.11026
[13] Haber, S., On a sequence of points of interest for numerical quadrature, Res. Nat. Bur. Standards Sect. B, 70, 127-134 (1966) · Zbl 0158.16002
[14] Hedlund, G. A.; Morse, M., Symbolic dynamics II, sturmian trajectories, Amer. J. Math., 62, 1-42 (1940)
[15] Hofbauer, F., On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math., 34, 213-237 (1979) · Zbl 0422.28015
[16] Hofbauer, F., The maximal measure for linear mod one transformations, J. London Math. Soc., 23, 92-112 (1981) · Zbl 0431.54025
[17] Hofbauer, F., The structure of piecewise monotonic transformations, Ergodic Theory Dynamical Systems, 1, 159-178 (1981) · Zbl 0474.28007
[18] Kuipers, L.; Niederreiter, H., Uniform Distribution of Sequences (1974), Wiley: Wiley New York · Zbl 0281.10001
[19] Milnor, J.; Thurston, W., On iterated maps of the interval, (Dold, A.; Eckmann, B., Lecture Notes in Mathematics. Lecture Notes in Mathematics, Dynamical Systems, Vol. 1342 (1988), Springer: Springer Berlin), 465-563 · Zbl 0664.58015
[20] Ostrowski, A., Bemerkung zur Theorie der Diophantischen Approximationen III, Abh. Math. Sem. Univ. Hamburg, 4, 224 (1926)
[21] Queffelec, M., Substitution Dynamical Systems, (Lecture Notes in Mathematics, Vol. 1424 (1987), Springer: Springer Berlin) · Zbl 0642.28013
[22] Ramshaw, L., On the discrepancy of the sequence formed by the multiples of an irrational number, J. Number Theory, 13, 138-175 (1981) · Zbl 0458.10035
[23] Schmidt, W., Irregularities of distributions VII, Acta Arith., 21, 45-50 (1972) · Zbl 0244.10035
[24] Takahashi, Y., Isomorphism of β-isomorphisms to Markov automorphisms, Osaka J. Math., 10, 175-184 (1973) · Zbl 0268.28006
[25] van der Corput, J. G., Verteilungsfunctionen I and II, Proc. Akad. Amsterdam, 39, 1058-1066 (1936) · Zbl 0013.05703
[26] Varga, R. S., Matrix Iterative Analysis, ((1962), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ), Automatic Comp. Series · Zbl 0133.08602
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