Expansion in $\text{SL}_2(\mathbb Z/q\mathbb Z)$ and Zaremba's conjecture

TL;DR

This paper develops an expansion theory for the group SL_2(Z/qZ) and uses it to confirm Zaremba's conjecture, advancing understanding of number theory and group expansion properties.

Contribution

It introduces a new expansion framework for SL_2(Z/qZ) and applies it to prove Zaremba's conjecture, a longstanding problem in number theory.

Findings

Established an expansion theory for SL_2(Z/qZ).

Confirmed Zaremba's conjecture using the new framework.

Abstract

We establish an expansion theory for $\text{SL}_2(\mathbb Z/q\mathbb Z)$. Incorporating this into a framework recently developed by Shkredov, we confirm Zaremba's conjecture.