no. 3
Irregularities of continuous distributions
Annales de l'Institut Fourier, Tome 39 (1989) no. 3, pp. 501-527

This paper deals with a continuous analogon to irregularities of point distributions. If a continuous fonction x:[0,1]→X where X is a compact body, is interpreted as a particle’s movement in time, then the discrepancy measures the difference between the particle’s stay in a proper subset and the volume of the subset. The essential part of this paper is to give lower bounds for the discrepancy in terms of the arc length of x(t), 0≤t≤1. Furthermore it is shown that these estimates are the best possible despite of logarithmic factors.

Cet article considère un analogue de l’irrégularité des ensembles discrets. Si X est un espace compact et x:[0,1]→X est une fonction continue (interprétée comme équation d’un mouvement), la discrépance mesure la différence entre le temps de séjour de x(t) dans certains sous-ensembles et le volume de ces sous-ensembles. Le but essentiel de cet article est de démontrer des estimations inférieures de la discrépance par une fonction de la longueur de l’arc de x(t), 0≤t≤1. De plus on démontre que les estimations sont optimales à des facteurs logarithmiques près.

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     author = {Drmota, Michael},
     title = {Irregularities of continuous distributions},
     journal = {Annales de l'Institut Fourier},
     pages = {501--527},
     year = {1989},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {3},
     doi = {10.5802/aif.1175},
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     zbl = {0665.10036},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1175/}
}
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Drmota, Michael. Irregularities of continuous distributions. Annales de l'Institut Fourier, Tome 39 (1989) no. 3, pp. 501-527. doi: 10.5802/aif.1175

[1] J. Beck, On a problem of K.F. Roth concerning irregularities of point distribution, Invent. Math., 74 (1980), 477-487. | Zbl | MR | EuDML

[2] J. Beck and W. Chen, Irregularities of distribution, Cambridge University Press, Cambridge-New York, 1987. | Zbl | MR

[3] M. Drmota, An optimal lower bound for the discrepancy of C-uniformly distributed functions modulo 1, Indag. Math. (1), 50 (1988), 21-28. | Zbl | MR

[4] M. Drmota, Untere Schranken für die C-Diskrepanz, Österr. Akad. Wiss., Math. Naturw. K1. SB II, 196 (1987), 107-117. | Zbl | MR

[5] M. Drmota and R.F. Tichy, C-uniform distribution on compact metric spaces, J. Math. Analysis & Appl. (1), 129 (1988), 284-292. | Zbl | MR

[6] M. Drmota and R.F. Tichy, Distribution properties of continuous curves, in: Théorie des nombres, Comptes Rendus de la Conférence internationale de Théorie des nombres tenue à l'Université Laval en 1987 (J.M. De Koninck, C. Levesque eds.), de Gruyter, Berlin - New York, 1989, 117-127. | Zbl | MR

[7] E. Hlawka, Über C-Gleichverteilung, Ann. Mat. Pura Appl. (IV), 49 (1960), 311-366. | Zbl | MR

[8] K.F. Roth, On irregularities of distribution, Mathematika, 1 (1954), 73-79. | Zbl | MR

[9] W.M. Schmidt, Irregularities of distribution IV, Invent. Math., 7 (1969), 55-82. | Zbl | MR | EuDML

[10] R.J. Taschner, The discrepancy of C-uniformly distributed multidimensional functions, J. Math. Analysis and Appl. (2), 78 (1980), 400-404. | Zbl | MR

[11] R.F. Tichy, Konvexe Mengen auf speziellen Mannigfaltigkeiten, Proc. 3. Kolloquium über Diskrete Geometrie (1985), 263-275. | Zbl

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