# The average number of integral points in orbits

Authors
Type
Preprint
Publication Date
Sep 14, 2015
Submission Date
Sep 02, 2015
Identifiers
arXiv ID: 1509.00752
Source
arXiv
Over a number field $K$, a celebrated result of Silverman states that if $\phi\in K(x)$ is a rational function whose second iterate is not a polynomial, the set of $S$-integral points in the orbit $\mathcal{O}_\phi(b)=\{\phi^n(b)\}_{n\geq0}$ is finite for all $b\in \mathbb{P}^1(K)$. In this paper, we show that if we vary $\phi$ and $b$ in a suitable family, the number of $S$-integral points of $\mathcal{O}_\phi(b)$ is absolutely bounded. In particular, if we fix $\phi\in K(x)$ and vary the base point $b\in \mathbb{P}^1(K)$, we show that $\#(\mathcal{O}_\phi(b)\cap\mathcal{O}_{K,S})$ is zero on average. Finally, we prove a zero-average result in general, assuming a standard height uniformity conjecture in arithmetic geometry, and prove it unconditionally over global function fields.