14 - Applications in Number Theory

Summary

Given the origins and motivation for studying the Assouad dimension, the applications in embedding theory are perhaps expected. In this chapter we discuss a collection of, perhaps more surprising, applications to several distinct problems in number theory. In Section 14.1 we consider the problem offinding arithmetic progressions inside sets of integers. Recall, for example, the Erdos conjecture on arithmetic progressions, of which the celebrated Green–Tao Theorem is a special case. It turns out that the Assouad dimension can be used to prove a weak `asymptotic' version of this conjecture. In Section 14.2 we discuss some problems in Diophantine approximation, the general area concerned with how well real numbers can be approximated by rationals. Here there is a well-developed connection between estimating the Hausdorff dimension of sets of badly approximable numbers and the lower dimension. Finally, in Section 14.3 we discuss an elegant use of the Assouad dimension in problems of definability of the integers due to Hieronymi and Miller.