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

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.

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.

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     author = {Drmota, Michael},
     title = {Irregularities of continuous distributions},
     journal = {Annales de l'Institut Fourier},
     pages = {501--527},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {3},
     year = {1989},
     doi = {10.5802/aif.1175},
     zbl = {0665.10036},
     mrnumber = {91b:11086},
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}
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1175/

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