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

Abstract

Building on Kowalski’s work on random walks on the groups $\mathrm{SL}(n,Z)$ and $\mathrm{Sp}(2g,Z)$, we consider similar problems (we try to estimate the probability with which, after $k$ steps, the matrix obtained has a characteristic polynomial with maximal Galois group or has no nonzero squares among its entries) for more general classes of sets: in $\mathrm{GL}(n,A)$, where $A$ is a subring of $Q$ containing $Z$ that we specify, we perform a random walk on the set of matrices with fixed determinant $D\in A^{\times }$. We also investigate the case where the set involved is any of the two left cosets of the special orthogonal group $\mathrm{SO}(n,m)(Z)$ with respect to the spinorial kernel $\Omega (n,m)(Z)$.