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On sparsity of integral points in orbits and correspondences with big pullbacks under iterates

Jorge Mello

TL;DR

The paper addresses the sparsity of $S$-integral points and algebraic points in forward orbits under multiple endomorphisms and under general correspondences on projective varieties over number fields. By combining the He–Ru Schmidt-type diophantine bounds with Ingram’s Correspondence Iterate Pullback (CIP) framework, it derives unconditional sparsity results in higher dimensions and for semigroup dynamics, extending prior one-variable results. It also establishes uniform sparsity for points of bounded degree and provides a path-space formulation for correspondences, including univariate corollaries that recover finiteness results for integers in dynamical correspondences. The results deepen the connection between height inequalities, Diophantine approximation, and arithmetic dynamics, with potential implications for broad conjectures in higher-dimensional Diophantine geometry.

Abstract

We prove new unconditional results of sparsity of integral points on orbits under many maps and correspondences in arbitrary dimensions, generalizing theorems of Yasufuku(2015) and others. The main ingredients are new diophantine approximation tools and recent constructions for correspondences due to Ingram (2011).

On sparsity of integral points in orbits and correspondences with big pullbacks under iterates

TL;DR

The paper addresses the sparsity of -integral points and algebraic points in forward orbits under multiple endomorphisms and under general correspondences on projective varieties over number fields. By combining the He–Ru Schmidt-type diophantine bounds with Ingram’s Correspondence Iterate Pullback (CIP) framework, it derives unconditional sparsity results in higher dimensions and for semigroup dynamics, extending prior one-variable results. It also establishes uniform sparsity for points of bounded degree and provides a path-space formulation for correspondences, including univariate corollaries that recover finiteness results for integers in dynamical correspondences. The results deepen the connection between height inequalities, Diophantine approximation, and arithmetic dynamics, with potential implications for broad conjectures in higher-dimensional Diophantine geometry.

Abstract

We prove new unconditional results of sparsity of integral points on orbits under many maps and correspondences in arbitrary dimensions, generalizing theorems of Yasufuku(2015) and others. The main ingredients are new diophantine approximation tools and recent constructions for correspondences due to Ingram (2011).
Paper Structure (8 sections, 19 theorems, 53 equations)

This paper contains 8 sections, 19 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on heights
  3. Tools from diophantine approximation
  4. Preliminaries on correspondences
  5. Proofs and results
  6. Sparsity of integral points in orbits
  7. Algebraic points of bounded degree in orbits
  8. Generalization to Correspondences

Key Result

Theorem 1.1

(Yasufuku)Y Let $\phi: \mathbb{P}^N \rightarrow \mathbb{P}^N$ be a morphism defined over $\mathbb{Q}$ of degree $d\geq 2$ and $D$ be a divisor of $\mathbb{P}^N$. Suppose there exists $m>0$ such that the normal-crossings subdivisor $D_{nc}^{(m)}$ of the pullback $(\phi^{(m)})^*(D)$ satisfies Vojta's

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 18 more