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

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 $\{{\sum }_{j=1}^m{c}_j{p}^{k_j{n}_j}:n_j\in {\mathbb{N}}_0\}$ ($c_j\in \mathbb{Q}$, $k_j\in {\mathbb{N}}_0$), there exist a split torus $X={\mathbb{G}}_m^k$ defined over $K=\overline{{\mathbb{F}}_p}(t)$, an endomorphism $\mathrm{\Phi }$ of $X$, $\alpha \in X(K)$ and a closed subvariety $V\subseteq X$ such that $\{n\in {\mathbb{N}}_0:{\mathrm{\Phi }}^n(\alpha )\in V(K)\}=S$.