Convex curves and a Poisson imitation of lattices

TL;DR

This paper investigates a probabilistic analogue of a classical open problem about convex curves and rational points, demonstrating that with probability, such curves can have infinitely many Poisson points as intensity grows.

Contribution

It introduces a Poisson-process-based approach to a longstanding open problem, showing the existence of convex curves with infinitely many points in a probabilistic setting.

Findings

Existence of convex curves with infinitely many Poisson points with probability 1.

Application of generalized affine length in the analysis.

Probabilistic analogue providing insights into the rational lattice problem.

Abstract

We solve a randomized version of the following open question: is there a strictly convex, bounded curve \gamma in the plane such that the number of rational points on \gamma, with denominator $n$, approaches infinity with $n$? Although this natural problem appears to be out of reach using current methods, we consider a probabilistic analogue using a spatial Poisson-process that simulates the refined rational lattice $\frac{1}{d} Z^2$, which we call $M_d$, for each natural number $d$. The main result here is that with probability 1 there exists a strictly convex, bounded curve \gamma such that the number of spatial Poisson points on \gamma, with intensity $d$, approaches infinity with $d$. The methods include the notion of a generalized affine length of a convex curve, defined by Petrov (2007).