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

The dimensions of integral points and holomorphic curves on the complements of hyperplanes