A sparsity result for the Dynamical Mordell-Lang Conjecture in positive characteristic

TL;DR

This paper provides a quantitative partial proof supporting the Dynamical Mordell-Lang Conjecture in positive characteristic, showing that the set of iterates hitting a subvariety has a controlled, sparsity-like structure.

Contribution

It establishes a new sparsity result for the set of points in the orbit of a semiabelian variety over a finite field that intersect a subvariety, advancing understanding of the DML conjecture in positive characteristic.

Findings

The set of iterates intersecting a subvariety is mostly a union of finitely many arithmetic progressions.

The remaining set has size bounded by a logarithmic function depending on the orbit length.

The result applies to semiabelian varieties over finite fields with regular self-maps.

Abstract

We prove a quantitative partial result in support of the Dynamical Mordell-Lang Conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field $K$ of characteristic $p$, given a semiabelian variety $X$ defined over a finite subfield of $K$ and endowed with a regular self-map $\Phi:X \longrightarrow X$ defined over $K$, given a point $\alpha\in X(K)$ and a subvariety $V\subseteq X$, then the set of all non-negative integers $n$ such that $\Phi^n(\alpha)\in V(K)$ is a union of finitely many arithmetic progressions along with a subset $S$ with the property that there exists a positive real number $A$ (depending only on $N$, $\Phi$, $\alpha$, $V$) such that for each positive integer $M$, we have $$\#\left\{n\in S\colon~ n\le M\right\}\le A\cdot \left(1+\log M\right)^{\dim V}.$$