Some Diophantine applications of the theory of group expansion

Summary

The recent years have seen considerable progress in the theory of expansion and spectral gaps for so-called thin groups (as opposed to the classical theory of arithmetic groups.) These developments rely in part on methods and results from the rather novel research area of “arithmetic combinatorics” and underlying are general principles such as the “sum-product theorem” in finite fields and “product theorems” in linear groups. These advances turned out to be of interest well beyond group theory and have applications to geometry, number theory, theoretical computer science and even mathematical physics. At this point, many aspects of the extensive story were already accounted for in several survey papers, such as [Bourgain 2010], the Bourbaki exposé of E. Kowalski [2012] and those of B. Green [2009] and A. Lubotzky [2012] based on AMS lectures. A discussion of the “ubiquity” of thin groups from a broader perspective appears in [Sarnak 2014] in the present volume.

We will focus here on two specific number-theoretic applications. The first relates to integral Apollonian circle packings (ACP for short) and the problem of a local/global principle for the curvatures, as proposed in [Graham et al. 2003] and [Sarnak 2011]. The other concerns progress towards Zaremba’s conjecture [1972] on continued fraction expansions of rationals and we will briefly review the paper [Bourgain and Kontorovich 2011]. Another exciting application of the theory of group expansion to finiteness in arithmetic geometry may be found in the work of J. Ellenberg, C. Hall and E. Kowalski [Ellenberg et al. 2012] but will not be discussed here. Neither will we get into the role of expansion to sieving theory (originating from [Bourgain et al. 2010a]) which triggered many of the later developments.

Our reference list is far from complete and strictly serves this exposé.