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

Abstract

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 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 [10] and expanded upon in [3] and [6].