Distribution of Lattice Points Visible from the Origin

Abstract:

Let Ω be a region in the plane which contains the origin, is star-shaped with respect to the origin and has a piecewise C 1 boundary. For each integer Q≥ 1, we consider the integer lattice points from which are visible from the origin and prove that the 1st consecutive spacing distribution of the angles formed with the origin exists. This is a probability measure supported on an interval [m Ω,∞), with m Ω >0. Its repartition function is explicitly expressed as the convolution between the square of the distance from origin function and a certain kernel.

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Received: 2 November 1999 / Accepted: 2 March 2000

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Boca, F., Cobeli, C. & Zaharescu, A. Distribution of Lattice Points Visible from the Origin. Commun. Math. Phys. 213, 433–470 (2000). https://doi.org/10.1007/s002200000250

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Keywords

  • Convolution
  • Probability Measure
  • Lattice Point
  • Origin Function
  • Spacing Distribution