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

Working over a base number field, 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 (). Here, f X X is a regular surjective self-map for X a geometrically irreducible projective variety over. Given a non-zero and effective f-quasi-polarizable Cartier divisor D on X and defined over, 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, Yasufuku: 2015, building on earlier work of Silverman 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 Ru: Vojta: 2016 and expanded upon in Grieve: points: bounded: degree and Grieve: qualitative: subspace.