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 Of(x) the forward f-orbit of a rational point x∈X(K). Here, f:X→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 Of(x). For the case of (D,S)-integral points, this result gives a sufficient condition for non-Zariski density of integral points in Of(x). 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.