The average number of integral points in orbits

TL;DR

This paper investigates the average number of $S$-integral points in orbits of rational functions over number fields, establishing finiteness, boundedness, and zero-average results under certain conditions and conjectures.

Contribution

It extends Silverman's finiteness result by showing uniform bounds and average zero results for $S$-integral points in orbits across families of rational functions.

Findings

Number of $S$-integral points in orbits is bounded when varying functions and points.

Average number of $S$-integral points in orbits is zero when fixing the function and varying the base point.

Conditional zero-average result assuming a height uniformity conjecture.

Abstract

Over a number field $K$, a celebrated result of Silverman states that if $\varphi(z)\in K(z)$ is a rational function whose second iterate is not a polynomial, the set of $S$-integral points in the orbit $\text{Orb}_\varphi(P)=\{\varphi^n(P)\}_{n\geq0}$ is finite for all $P\in \mathbb{P}^1(K)$. In this paper, we show that if we vary $\varphi$ and $P$ in a suitable family, the number of $S$-integral points in $\text{Orb}_\varphi(P)$ is absolutely bounded. In particular, if we fix $\varphi$ and vary the basepoint $P\in \mathbb{P}^1(K)$, then we show that $\#(\text{Orb}_\varphi(P)\cap\mathcal{O}_{K,S})$ is zero on average. Finally, we prove a zero-average result in general, assuming a standard height uniformity conjecture in arithmetic geometry.