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

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 Cq^{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.