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On the number of lattice points between two enlarged and randomly shifted, copies of an oval
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  • Published: June 1994

On the number of lattice points between two enlarged and randomly shifted, copies of an oval

  • Zheming Cheng1,
  • Joel L. Lebowitz1,2 &
  • Péter Major3 

Probability Theory and Related Fields volume 100, pages 253–268 (1994)Cite this article

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Summary

Let A be an oval with a nice boundary in ℝ2,R a large positive number,c>0 some fixed number and α a uniformly distributed random vector in the unit square [0,1]2. We are interested in the number of lattice points in the shifted annular region consisting of the difference of the sets {(R+c/R)A−α} and {(R−c/R)A−α}. We prove that whenR tends to infinity, the expectation and the variance of this random variable tend to 4c times the area of the set A, i.e. to the area of the domain where we are counting the number of lattice points. This is consistent with computer studies in the case of a circle or an ellipse which indicate that the distribution of this random variable tends to the Poisson law. We also make some comments about possible generalizations.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    Zheming Cheng & Joel L. Lebowitz

  2. Department of Physics, Rutgers University, 08903, New Brunswick, NJ, USA

    Joel L. Lebowitz

  3. Mathematical Institute of the Hungarian Academy of Sciences, P.O.B. 127, H-1364, Budapest, Hungary

    Péter Major

Authors
  1. Zheming Cheng
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  2. Joel L. Lebowitz
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  3. Péter Major
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Cheng, Z., Lebowitz, J.L. & Major, P. On the number of lattice points between two enlarged and randomly shifted, copies of an oval. Probab. Th. Rel. Fields 100, 253–268 (1994). https://doi.org/10.1007/BF01199268

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  • Received: 03 January 1994

  • Issue Date: June 1994

  • DOI: https://doi.org/10.1007/BF01199268

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Mathematics Subject Classifications (1991)

  • 60F5
  • 60K4
  • 11K06
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