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Combinatorial Applications of the Subspace Theorem
The Subspace Theorem is a powerful tool in number theory. It has appeared in
various forms and been adapted and improved over time. It's 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 sum-product estimates and bounds on unit distances. The
presentation will be from the point of view of a discrete mathematician. We
will state a number of variants of the Subspace Theorem below but we will not
prove any of them as the proofs are beyond the scope of this work. However we
will give a proof of a simplified special case of the Subspace Theorem which is
still very useful for many problems in discrete mathematics
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