The Dimensions of Integral Points and Holomorphic Curves on the Complements of Hyperplanes

In this article we completely determine the possible dimensions of integral points and holomorphic curves on the complement of a union of hyperplanes in projective space. Our main theorems generalize a result of Evertse and Gyory, who determined when all sets of integral points (over all number fields) on the complement of a union of hyperplanes are finite, and a result of Ru, who determined when all holomorphic maps to the complement of a union of hyperplanes are constant. The main tools used are the S-unit lemma and its analytic analogue, Borel's lemma.