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}.