On non-Zariski density of $(D,S)$-integral points in forward orbits and the Subspace Theorem

TL;DR

This paper establishes conditions under which $(D,S)$-integral points in forward orbits of rational maps on algebraic varieties are not Zariski dense, using the Subspace Theorem and dynamics of divisors.

Contribution

It provides an unconditional criterion for non-Zariski density of integral points in orbits, expanding previous work by Yasufuku and Silverman with new dynamical and arithmetic insights.

Findings

Provides sufficient conditions for non-Zariski density of integral points in orbits.

Utilizes the Subspace Theorem in a generalized form by Ru and Vojta.

Builds on and extends earlier results by Silverman and Yasufuku.

Abstract

Working over a base number field $\KK$, we study the attractive question of Zariski non-density for $(D,S)$-integral points in $\mathrm{O}_f(x)$ the forward $f$-orbit of a rational point $x \in X(\KK)$. Here, $f \colon X \rightarrow X$ is a regular surjective self-map for $X$ a geometrically irreducible projective variety over $\KK$. Given a non-zero and effective $f$-quasi-polarizable Cartier divisor $D$ on $X$ and defined over $\KK$, our main result gives a sufficient condition, that is formulated in terms of the $f$-dynamics of $D$, for non-Zariski density of certain dynamically defined subsets of $\mathrm{O}_f(x)$. For the case of $(D,S)$-integral points, this result gives a sufficient condition for non-Zariski density of integral points in $\mathrm{O}_f(x)$. Our approach expands on that of Yasufuku, \cite{Yasufuku:2015}, building on earlier work of Silverman \cite{Silverman:1993}. Our main result gives an unconditional form of the main results of loc.~cit.; the key arithmetic input to our main theorem is the Subspace Theorem of Schmidt in the generalized form that has been given by Ru and Vojta in \cite{Ru:Vojta:2016} and expanded upon in \cite{Grieve:points:bounded:degree} and \cite{Grieve:qualitative:subspace}.