Convergence in total variation of an affine random recursion in [0,p)^k to a uniform random vector

We study the rate of convergence of the Markov chain $\mathbf{X}_{n+1}=A\mathbf{X}_{n}+\mathbf{B}_{n}$ (mod $p$), where $A$ is an integer matrix with nonzero eigenvalues, $p$ is real and positive, and $\left\{ \mathbf{B}_{n}\right\} $ is a sequence of independent and identically distributed real random vectors. With some hypotheses on the law of $\mathbf{B}_{n}$, the sequence $\left\{ \mathbf{X}_{n}\right\} $ converges to a random vector uniformly distributed in $[0,p)^{k}$. The rate of convergence is geometric and depends on $A$, $p$, $k$, and the distribution of $\mathbf{B}_{n}$. Moreover, if $A$ has an eigenvalue that is a root of $1$, then $n=O\left( p^{2}\right) $ steps are necessary to have $\mathbf{X}_{n}$ sampling from a nearly uniform law.