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

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}.