Applications of Thin Orbits

TL;DR

This paper reviews the theory of thin orbits, their connections to number theory and geometry, and discusses a unifying conjecture that links various problems in the field, along with partial progress on this conjecture.

Contribution

It introduces a unifying 'Local-Global' Conjecture for thin orbits that connects multiple problems in dynamics and number theory, and discusses partial progress.

Findings

Partial progress on the 'Local-Global' Conjecture.

Connections established between thin orbits and classical problems.

Insights into equidistribution and reduction theory.

Abstract

This text is based on a series of three expository lectures on a variety of topics related to "thin orbits," as delivered at Durham University's Easter School on "Dynamics and Analytic Number Theory" in April 2014. The first lecture reviews closed geodesics on the modular surface and the reduction theory of binary quadratic forms before discussing Duke's equidistribution theorem (for indefinite classes). The second lecture exposits three quite different but (it turns out) not unrelated problems, due to Einsiedler-Lindenstrauss-Michel-Venkatesh, McMullen, and Zaremba. The third lecture reformulates these in terms of the aforementioned thin orbits, and shows how all three would follow from a single "Local-Global" Conjecture of Bourgain and the author. We also describe some partial progress on the conjecture, which has lead to some results on the original problems.