Angular distribution toward the points of the neighbor‐flips modular curve seen by a fast moving observer

Abstract Let be a fixed non‐zero integer. For every and every prime , consider the angles between rays from an observer located at the point on the real axis toward the set of all integral solutions of the equation in the square , where . This set of points can be seen as a generic model for any target set with points randomly distributed on the integer coordinates of a square, in which, apart from a small number of exceptions, exactly one point lies above any abscissa. We prove the existence of the limiting gap distribution for this set of angles as , providing explicit formulas for the corresponding density function, which turns out to be independent of . The resulted gap distribution function shows the existence of a sequence of threshold points between which the distribution of seen angles has different shapes. This provides a tool of reference in guiding the observer, which allows one to find and control the position relative to the universe of observed points.