A converse of dynamical Mordell--Lang conjecture in positive characteristic

TL;DR

This work proves the converse of the dynamical Mordell-Lang conjecture in positive characteristic: for any $S\subseteq \mathbb{N}_0$ that is a finite union of arithmetic progressions and $p$-sets of the form $\{ \sum_{j=1}^m c_j p^{k_j n_j} : n_j\in \mathbb{N}_0\}$, there exists a split torus $X=\mathbb{G}_m^k$, an endomorphism $\Phi$, a point $\alpha$, and a subvariety $V$ defined over $K=\overline{\mathbb{F}_p}(t)$ such that $S=\{ n: \Phi^n(\alpha)\in V(K) \}$. The proof proceeds by reducing to a two-factor setting, decomposing $p$-sets into smaller blocks $B(q; c_1,\dots,c_m; 1,\dots,1)$ and applying a key dynamical-DML result to each piece, then assembling via closure lemmas. An inductive argument on the number of summands $m$ shows all blocks are $\overline{\mathbb{F}_p}(t)$-DML sets over split tori, yielding the full converse. This advances arithmetic dynamics over function fields in characteristic $p$ and provides a constructive realization framework for prescribed combinatorial sets as DML hitting sets.

Abstract

In this paper, we prove the converse of the dynamical Mordell--Lang conjecture in positive characteristic: For every subset $S \subseteq \mathbb{N}_0$ which is a union of finitely many arithmetic progressions along with finitely many $p$-sets of the form $\left \{ \sum_{j=1}^{m} c_j p^{k_jn_j} : n_j \in \mathbb{N}_0 \right \}$ ($c_j \in \mathbb{Q}$, $k_j \in \mathbb{N}_0$), there exist a split torus $X = \mathbb{G}_m^k$ defined over $K=\overline{\mathbb{F}_p}(t)$, an endomorphism $Φ$ of $X$, $α\in X(K)$ and a closed subvariety $V \subseteq X$ such that $\left \{ n \in \mathbb{N}_0 : Φ^n(α) \in V(K) \right \} = S$.

Theorems & Definitions (20)

• Conjecture 1.1
• Theorem 1.2
• Example 1.3
• Conjecture 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof