Lifting all elements in ${\mathrm{SL}}_n(\mathbb{Z}/q\mathbb{Z})$

Abstract

We show that every element of ${\mathrm{SL}}_n(\mathbb{Z}/q\mathbb{Z})$ can be lifted to an element of ${\mathrm{SL}}_n(\mathbb{Z})$ of norm at most $C{q}^2\log q$, while there exists an element such that every lift of it is of norm at least $q^{2+o(1)}$. This should be compared to the recent result that almost every element has a lift of norm bounded by $q^{1+1/n+o(1)}$. The main step in the proof is showing that for every $q$, there is a small element in $(\mathbb{Z}/q\mathbb{Z}{)}^{\times }$ with a large $n$-th root, which is a result of independent interest. In the proof we use tools from additive combinatorics including Bohr sets.