$S$-Integral Points in Orbits on $\mathbb{P}^1$

Technical Summary: S-Integral Points in Orbits on $\mathbb{P}^1$

Problem Statement
Let $K$ be a number field and $S$ a finite set of places containing all archimedean places. Consider a rational map $\phi: \mathbb{P}^1 \to \mathbb{P}^1$ of degree $d \ge 2$ defined over $K$. For a non-preperiodic point $\alpha \in \mathbb{P}^1(K)$ and a non-exceptional point $\beta \in \mathbb{P}^1(K)$, the paper investigates the number of points in the forward orbit $\{\phi^n(\alpha)\}_{n \ge 0}$ that are $S$-integral relative to $\beta$. Two points $x, y$ are $S$-integral relative to each other if their reductions are distinct modulo $v$ for all places $v \notin S$.

While Silverman [Sil93] established the finiteness of such points, quantitative bounds have been a subject of study. Previous work by Hsia and Silverman [HS11] provided a bound of the form $c \cdot 4^{|S|}$, and Krieger et al. [KLS+15] established an exponential bound $c^{|S|}$. The primary objective of this paper is to improve the dependence on the size of the set $S$, $|S|$, from exponential to superlinear.

Methodology
The proof strategy follows the dynamical framework established by Silverman [Sil93] and Hsia–Silverman [HS11], utilizing the canonical height $\hat{h}_\phi$ and the Weil height $h$. The core approach involves relating the $S$-integrality condition to a quantitative version of Roth's theorem.

1. Reduction and Normalization: The authors first reduce the general case to a scenario where the canonical height of $\alpha$ is sufficiently large relative to the height of the map $\phi$ and the point $\beta$. This involves iterating $\alpha$ and applying affine conjugations to ensure $\infty$ does not appear in the relevant orbits or preimages of $\beta$.
2. Non-Superattracting Case: When $\beta$ is not part of a superattracting cycle, the proof leverages the inverse function theorem for rational maps. By analyzing the preimages $\phi^{-N}(\beta)$ for a carefully chosen integer $N$ (dependent on $|S|$), the authors show that if $\phi^n(\alpha)$ is $S$-integral, then $\phi^{n-N}(\alpha)$ must be exceptionally close to a specific preimage $\beta_i$ at some place $v \in S$.
   • The authors employ a pigeonhole argument combined with a combinatorial analysis of the distribution of "exceptional" places. They define subsets of indices based on how many places satisfy a specific height inequality derived from Roth's theorem.
   • Crucially, instead of applying Roth's theorem over the entire set $S$ simultaneously (which yields exponential bounds), the authors apply a quantitative version of Roth's theorem (Theorem 2.1, due to Gross [Gro90]) to a single place $v$ at a time. This allows for a tighter bound on the number of exceptions.
3. Superattracting Case: When $\beta$ lies in a superattracting cycle, the multiplicity of the preimages grows, rendering the inverse function approach less effective. Instead, the authors analyze the growth of the local distance $|\phi^n(\alpha) - \beta|_v$ directly. Using Taylor series expansions and properties of superattracting cycles, they demonstrate that the distance decays at a rate determined by the multiplicity $e$. By comparing this decay rate with the growth of the global height $h(\phi^n(\alpha))$, they derive a contradiction for sufficiently large $n$ without invoking Roth's theorem, relying instead on height inequalities and the product formula.
4. Uniformity: For polynomials with bounded bad reduction, the paper utilizes results by Looper [Loo19] regarding the uniform lower bound of canonical heights. By conjugating the polynomial to a normal form and utilizing the concept of "quasi-$(S, \epsilon)$-integrality," the authors control the effect of conjugation on $S$-integrality to establish uniform bounds.

Key Results

• Theorem 1.1 (Main Bound): For a fixed number field $K$ and rational map $\phi$, there exists a constant $c_1$ such that for any non-preperiodic $\alpha$ and non-exceptional $\beta$, the number of $S$-integral points in the orbit is bounded by:
  $$|\{n \ge 0 \mid \phi^n(\alpha) \text{ is } S\text{-integral relative to } \beta\}| \le c_1 |S| (\log |S| + 1)^5$$
  This represents a significant improvement over the previous exponential bounds in $|S|$.

• Theorem 1.2 (Explicit Dependence): A more precise bound is provided that explicitly tracks the dependence on the height of the map and the canonical height of $\alpha$:
  $$\text{Count} \le c_2 |S| (\log |S| + 1)^6 + \log_d^+ \left( \frac{h(\phi) + 1}{\hat{h}_\phi(\alpha)} \right)$$
  The paper notes that the dependency on the ratio $h(\phi)/\hat{h}_\phi(\alpha)$ is necessary and cannot be removed in the general rational map case (Proposition 6.1).

• Theorem 1.3 (Uniformity for Polynomials): If $\phi(z)$ is a polynomial of degree $d$ with at most $s$ places of bad reduction (and $\phi(z) \neq cz^d$), there exists a constant $c_3$ depending on $[K:\mathbb{Q}], d, s, |S|$ such that the number of $S$-units in the orbit is uniformly bounded by $c_3$. This extends the results of Krieger et al. [KLS+15] to polynomials with bounded bad reduction.

Significance and Claims
The paper claims to provide the first superlinear upper bound on the number of $S$-integral points in a dynamical orbit, improving upon the exponential bounds previously known. The improvement is achieved by refining the application of Roth's theorem, moving from a global application over $S$ to a localized application over single places, coupled with a combinatorial optimization of the preimage structure.

The authors explicitly state that the dependency on the height ratio $h(\phi)/\hat{h}_\phi(\alpha)$ in the general case is unavoidable, as demonstrated by a constructed counterexample. However, for the specific class of polynomials with bounded bad reduction, they successfully remove this dependency to achieve a uniform bound, addressing a question raised in [KLS+15]. The results are presented as quantitative refinements of existing finiteness theorems in arithmetic dynamics, relying on established tools like Roth's theorem and height inequalities rather than introducing new deep conjectures.