Angular distribution towards the points of the neighbor-flips modular curve seen by a fast moving observer

TL;DR

The paper investigates the angular gap distribution of points on the neighbor-flips modular curve $\mathcal{A}(p,h)$ as seen from a moving observer at $P_{t,J}=(-tJ^2,0)$, with $J=(p-1)/2$, and proves the existence of a limiting gap distribution as $p\to\infty$ along primes with $p>|h|$. The authors develop a framework based on the distribution of rational functions modulo a prime, leveraging Weil-type exponential-sum bounds and a multidimensional counting lemma to show that the joint distribution of the relevant rational functions is equidistributed with a controllable error, enabling the explicit computation of the limit. They show that the limiting gap distribution $G_{t,\mathcal{P}}(\lambda)$ exists and equals $2\mu(\Omega(t,\lambda))$, is independent of $h$, and, for $t\ge 2$, admits a closed-form, piecewise description; the density $g_{t,\mathcal{P}}(\lambda)=-\partial_λG_{t,\mathcal{P}}(λ)$ is likewise given. The results connect modular-curve point distributions, lattice-point statistics, and equidistribution phenomena for rational functions modulo primes, with insights into how the observer’s distance (via $t$) shapes the observed gap structure, including asymptotic exponential behavior as $t\to 1/J$.

Abstract

Let $h$ be a fixed non-zero integer. For every $t\in \mathbb{R}_+$ and every prime $p$, consider the angles between rays from an observer located at the point $(-tJ_p^2,0)$ on the real axis towards the set of all integral solutions $(x,y)$ of the equation $y^{-1}-x^{-1}\equiv h \pmod{p}$ in the square $[-J_p,J_p]^2$, where $J_p=(p-1)/2$. We prove the existence of the limiting gap distribution for this set of angles as $p\rightarrow \infty$, providing explicit formulas for the corresponding density function, which turns out to be independent of $h$.

Figures (11)

• Figure 1: The points of $\mathcal{A}(359,26)$, seen by an observer placed at $(-tJ_{359}^2,0)$.
• Figure 2: Graphs of $G_{t,q,h}(\lambda)$ computed numerically and thus referred to as $G^*_{t,q,h}(\lambda)$. Five graphs are shown in each image, for $q=7879, 7880,7881,7882$ and $7883$. Notice that $q=7879$ and $7883$ are both prime and the difference between the corresponding graphs is indiscernible. In the image on the left, $h=2$ and $t=1.5$, while in the one on the right, $h=64$ and $t=1.25$.
• Figure 3: The graph of the limit $G_{t,\mathcal{P}}(\lambda)$ for $t\ge 2$. In the image on the left-hand side, the graph of $G^*_{t,q,h} (\lambda)$, calculated numerically, is overlaid with dots. Here $t = 2.76, p=10\,007$ and $h=1$. In the image on the right-hand side, the graph of $G_{t,\mathcal{P}}(\lambda)$ is shown for four different values of $t$, that is, $t=2,2.76,5$ and $22$. The distinct expressions that $G_{t,\mathcal{P}}(\lambda)$ takes for $\lambda \ge 0$ are indicated by different shades of the corresponding subgraph regions.
• Figure 4: The graph of the limit density $g_{t,\mathcal{P}}(\lambda)$ for $t\ge 2$. In the image on the left-hand side, $t = 2.76$, whereas in the image on the right-hand side, the graph of $g_{t,\mathcal{P}}(\lambda)$ is is shown for $t=2,2.76,5$ and $22$.
• Figure 5: The graph of $G_{t,\mathcal{P}}(\lambda)$ for $t=1.45$ (left) and $t=1.12$ (right). In each case, the dotted graph of $G^*_{t,q,h}(\lambda)$ calculated numerically with $q=8009$ and $h=1$ is overlaid on top. The shaded regions of the subgraphs indicate the regions where $G_{t,\mathcal{P}}(\lambda)$ has different expressions.

Theorems & Definitions (8)

• Theorem 1
• Corollary 1
• Lemma 1: Zah2003
• Lemma 2
• proof
• Corollary 2
• proof
• Remark 1