Sets of block structure and discrepancy estimates
Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 337-349.

Soient 𝐱=(x n ) n une suite d'éléments d'un ensemble fini M et 𝐟 = (f n ) n une suite d'applications f n :MM. Quelle information sur 𝐱 et 𝐟 permet d'obtenir des estimations de la discrépance de la suite 𝐟(𝐱)=(f n (x n )) n ? Nous donnons dans cet article des réponses à cette question, en utilisant un résultat qualitatif récent.

Given a sequence 𝐱=(x n ) n on the finite set M and a sequence 𝐟 = (f n ) n of maps f n :MM. Which information about 𝐱 and 𝐟 is suitable for getting estimates for the discrepancy of the sequence 𝐟(𝐱)=(f n (x n )) n ? The paper's object is, using a recent qualitative result, to give answers to this question.

@article{JTNB_1997__9_2_337_0,
     author = {Winkler, Reinhard},
     title = {Sets of block structure and discrepancy estimates},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {337--349},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {2},
     year = {1997},
     doi = {10.5802/jtnb.206},
     zbl = {0899.11036},
     mrnumber = {1617402},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_1997__9_2_337_0/}
}
Reinhard Winkler. Sets of block structure and discrepancy estimates. Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 337-349. doi : 10.5802/jtnb.206. https://jtnb.centre-mersenne.org/item/JTNB_1997__9_2_337_0/

[D-T] M. Drmota and R.F. Tichy. Sequences, discrepancy and applications. Springer, Lecture Notes in Mathematics 1651 (1997). | MR 1470456 | Zbl 0877.11043

[G] M. Goldstern. The complexity of uniform distribution. Math. Slovaca 44.5 (1994), 491-500. | | MR 1338422 | Zbl 0820.03031

[H] E. Hlawka. Theorie der Gleichverteilung. Bibl. Inst. Mannheim - Wien- Zürich, 1979. | MR 542905 | Zbl 0406.10001

[Ki-Li] H. Ki and T. Linton. Normal numbers and subsets of N with given densities. Fund. Math. 144 (1994), 163-179. | | MR 1273694 | Zbl 0809.04001

[Kui-N] L. Kuipers and H. Niederreiter. Uniform distribution of sequences. John Wiley and Sons, New York, 1974. | MR 419394 | Zbl 0281.10001

[Lo] V. Losert. Almost constant sequences of transformations. Mh. Math. 85 (1978), 105-113. | | MR 476678 | Zbl 0381.28006

[Lo-Ri] V. Losert and H. Rindler. Almost constant sequences. Soc. Math. de France Astérisque 61 (1979), 133-143. | MR 556670 | Zbl 0404.22004

[Rau] G. Rauzy. Etude de quelques ensembles de fonctions définis par des proprietés de moyenne. Théorie des Nombres Univ. de Bordeaux 1972/73, Exp. 20. | | MR 396463 | Zbl 0293.10018

[Ri] H. Rindler. Fast konstante Folgen. Acta Arith. 35.2 (1979), 189-193. | | MR 547674 | Zbl 0335.10051

[W] R. Winkler. Distribution preserving sequences of maps and almost constant sequences on finite sets. To appear in Mh. Math. | | MR 1639383 | Zbl 0908.11035