Irregularities of continuous distributions
Annales de l'Institut Fourier, Volume 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), 0t1. 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), 0t1. De plus on démontre que les estimations sont optimales à des facteurs logarithmiques près.

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     title = {Irregularities of continuous distributions},
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     pages = {501--527},
     publisher = {Institut Fourier},
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Drmota, Michael. Irregularities of continuous distributions. Annales de l'Institut Fourier, Volume 39 (1989) no. 3, pp. 501-527. doi : 10.5802/aif.1175. http://archive.numdam.org/articles/10.5802/aif.1175/

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