Combinatorial applications of the subspace theorem

Abstract

The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. Its applications include diophantine approximation, results about integral points on algebraic curves and the construction of transcendental numbers. But its usefulness extends beyond the realms of number theory. Other applications of the Subspace Theorem include linear recurrence sequences and finite automata. In fact, these structures are closely related to each other and the construction of transcendental numbers. The Subspace Theorem also has a number of remarkable combinatorial applications. The purpose of this paper is to give a survey of some of these applications including bounds on unit distances, sum-product estimates and a result about the structure of complex lines. The presentation will be from the point of view of a discrete mathematician. We will state a number of variants and a corollary of the Subspace Theorem and give a proof of a simplified special case of the corollary which is still very useful for many problems in discrete mathematics.