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

Jack Anderson, Florin P. Boca, Cristian Cobeli and Alexandru Zaharescu

Mathematische Nachrichten, 2025, vol. 298, issue 5, 1617-1632

Abstract: Let h$h$ be a fixed non‐zero integer. For every t∈R+$t\in \mathbb {R}_+$ and every prime p$p$, consider the angles between rays from an observer located at the point (−tJp2,0)$(-tJ_p^2,0)$ on the real axis toward the set of all integral solutions (x,y)$(x,y)$ of the equation y−1−x−1≡hmodp$y^{-1}-x^{-1}\equiv h \left(\mathrm{ mod\;}p\right)$ in the square [−Jp,Jp]2$[-J_p,J_p]^2$, where Jp=(p−1)/2$J_p=(p-1)/2$. 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 p→∞$p\rightarrow \infty$, providing explicit formulas for the corresponding density function, which turns out to be independent of h$h$. 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.

Date: 2025
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