Integral points of bounded degree on the projective line and in dynamical orbits

TL;DR

This paper proves that sets of bounded degree integral points on the projective line are sparse and applies this to show that algebraic integers in the orbits of certain rational functions are rare, with bounded and zero average counts.

Contribution

It establishes the sparsity of bounded degree integral points on the projective line and applies this to bound and analyze algebraic integers in dynamical orbits.

Findings

Sets of bounded degree integral points have density zero.

Number of algebraic integers in orbits is bounded.

Average number of algebraic integers in orbits is zero.

Abstract

Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $\varphi(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $\varphi^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.