Irregularities of continuous distributions

Michael Drmota

Annales de l'institut Fourier (1989)

  • Volume: 39, Issue: 3, page 501-527
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Drmota, Michael. "Irregularities of continuous distributions." Annales de l'institut Fourier 39.3 (1989): 501-527. <http://eudml.org/doc/74839>.

@article{Drmota1989,
abstract = {This paper deals with a continuous analogon to irregularities of point distributions. If a continuous fonction $x: [0,1]\rightarrow X$ 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 $x(t)$, $0\le t\le 1$. Furthermore it is shown that these estimates are the best possible despite of logarithmic factors.},
author = {Drmota, Michael},
journal = {Annales de l'institut Fourier},
keywords = {uniform distribution; uniformly distributed functions; irregularities of point distributions; discrepancy; Beck's Fourier transform method},
language = {eng},
number = {3},
pages = {501-527},
publisher = {Association des Annales de l'Institut Fourier},
title = {Irregularities of continuous distributions},
url = {http://eudml.org/doc/74839},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Drmota, Michael
TI - Irregularities of continuous distributions
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 3
SP - 501
EP - 527
AB - This paper deals with a continuous analogon to irregularities of point distributions. If a continuous fonction $x: [0,1]\rightarrow X$ 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 $x(t)$, $0\le t\le 1$. Furthermore it is shown that these estimates are the best possible despite of logarithmic factors.
LA - eng
KW - uniform distribution; uniformly distributed functions; irregularities of point distributions; discrepancy; Beck's Fourier transform method
UR - http://eudml.org/doc/74839
ER -

References

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  1. [1] J. BECK, On a problem of K.F. Roth concerning irregularities of point distribution, Invent. Math., 74 (1980), 477-487. Zbl0528.10037MR85g:11063
  2. [2] J. BECK and W. CHEN, Irregularities of distribution, Cambridge University Press, Cambridge-New York, 1987. Zbl0617.10039MR88m:11061
  3. [3] M. DRMOTA, An optimal lower bound for the discrepancy of C-uniformly distributed functions modulo 1, Indag. Math. (1), 50 (1988), 21-28. Zbl0642.10048MR89c:11114
  4. [4] M. DRMOTA, Untere Schranken für die C-Diskrepanz, Österr. Akad. Wiss., Math. Naturw. K1. SB II, 196 (1987), 107-117. Zbl0658.10054MR90a:11093
  5. [5] M. DRMOTA and R.F. TICHY, C-uniform distribution on compact metric spaces, J. Math. Analysis & Appl. (1), 129 (1988), 284-292. Zbl0639.10034MR89a:11078
  6. [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. Zbl0679.10041MR90j:11081
  7. [7] E. HLAWKA, Über C-Gleichverteilung, Ann. Mat. Pura Appl. (IV), 49 (1960), 311-366. Zbl0091.04703MR22 #7996
  8. [8] K.F. ROTH, On irregularities of distribution, Mathematika, 1 (1954), 73-79. Zbl0057.28604MR16,575c
  9. [9] W.M. SCHMIDT, Irregularities of distribution IV, Invent. Math., 7 (1969), 55-82. Zbl0172.06402MR39 #6838
  10. [10] R.J. TASCHNER, The discrepancy of C-uniformly distributed multidimensional functions, J. Math. Analysis and Appl. (2), 78 (1980), 400-404. Zbl0468.10029MR82h:10064
  11. [11] R.F. TICHY, Konvexe Mengen auf speziellen Mannigfaltigkeiten, Proc. 3. Kolloquium über Diskrete Geometrie (1985), 263-275. Zbl0554.52001

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