The large sieve and random walks on left cosets of arithmetic groups

TL;DR

This paper uses a generalized large sieve to show that typical algebraic properties hold for most matrices in certain groups, such as irreducibility and absence of squares, with explicit quantitative results.

Contribution

It applies E. Kowalski's large sieve generalization to prove that expected algebraic properties are prevalent among matrices in specific arithmetic groups.

Findings

Most matrices have irreducible characteristic polynomials.

Matrices typically do not have squares among their entries.

Results are explicit and apply to matrices with fixed determinant or spinor norm.

Abstract

Applying E. Kowalski's recent generalization of the large sieve we prove that certain properties expected to be typical (irreducibility of the characteristic polynomial, absence of squares among the matrix coefficients...) are indeed verified by most (in a very explicit sense) of the elements of GL(n,A) with fixed determinant (where A is an intermediate ring between Z and Q that we specify) or by (special) orthogonal matrices with integral entries and fixed spinor norm.