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Poisson law for the number of lattice points in a random strip with finite area
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  • Published: December 1992

Poisson law for the number of lattice points in a random strip with finite area

  • Péter Major1 

Probability Theory and Related Fields volume 92, pages 423–464 (1992)Cite this article

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  • 14 Citations

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Summary

Let a smooth curve be given by a functionr=f(ϕ) in polar coordinate system in the plane, and letR be a uniformly distributed random variable on the interval [a 1 L, a 2 L] with somea 2>a 1>0 and a largeL>0. Ya. G. Sinai has conjectured that given some real numbersc 2>c 1, the number of lattice points in the domain between the curves\(\left( {R + \frac{{c_1 }}{R}} \right)\) and\(\left( {R + \frac{{c_2 }}{R}} \right)\) is asymptotically Poisson distributed for “good” functionsf(·). We cannot prove this conjecture, but we show that if a probability measure with some nice properties is given on the space of smooth functions, then almost all functions with respect to this measure satisfy Sinai's conjecture. This is an improvement of an earlier result of Sinai [9], and actually the proof also contains many ideas of that paper.

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Authors and Affiliations

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

    Péter Major

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  1. Péter Major
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This article was processed by the author using the Springer-Verlag TEX ProbTh macro package 1991.

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Major, P. Poisson law for the number of lattice points in a random strip with finite area. Probab. Th. Rel. Fields 92, 423–464 (1992). https://doi.org/10.1007/BF01274263

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  • Received: 19 March 1991

  • Revised: 25 November 1991

  • Issue Date: December 1992

  • DOI: https://doi.org/10.1007/BF01274263

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Keywords

  • Coordinate System
  • Stochastic Process
  • Probability Measure
  • Smooth Function
  • Probability Theory
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