Work

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

Year: 2026

Type: article

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 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.

Source: Journal of the European Mathematical Society

DOI: 10.4171/jems/1790

Authors: Amitay Kamber, Péter P. Varjú

Institution: University of Cambridge

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Royal Society, European Commission

Award: G434337789