From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits

TL;DR

This paper explores the interconnected local-global problems in arithmetic related to thin orbits, such as Apollonian gaskets and Zaremba's Conjecture, highlighting recent progress using diverse modern mathematical techniques.

Contribution

It unifies various arithmetic problems involving thin orbits and discusses recent advances achieved through interdisciplinary methods.

Findings

Progress on local-global problems for Apollonian gaskets

Partial results towards Zaremba's Conjecture

Application of harmonic analysis, algebra, and dynamics

Abstract

We discuss a number of naturally arising problems in arithmetic, culled from completely unrelated sources, which turn out to have a common formulation involving "thin" orbits. These include the local-global problem for integral Apollonian gaskets and Zaremba's Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above.