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

Working over a base number field K, we study the attractive question of Zariski non-density for $(D,S)$-integral points in $O_f\left(x\right)$ the forward f-orbit of a rational point $x\in X\left(K\right)$. Here, $f:X\to X$ is a regular surjective self-map for X a geometrically irreducible projective variety over K. Given a non-zero and effective f-quasi-polarizable Cartier divisor D on X and defined over K, 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 $O_f\left(x\right)$. For the case of $(D,S)$-integral points, this result gives a sufficient condition for non-Zariski density of integral points in $O_f\left(x\right)$. Our approach expands on that of Yasufuku, [13], building on earlier work of Silverman [11]. Our main result gives an unconditional form of the main results of [13]; 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 [10] and expanded upon in [3] and [6].