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 1^st 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.