Typical self-affine sets with non-empty interior

Abstract: Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $|T_i| <1/2$ for $1\leq i\leq m$. We provide some sufficient conditions on these matrices such that the self-affine set generated by the iterated function system ${T_ix+a_i}_{i=1}^m$ on $\Bbb R^d$ has non-empty interior for almost all $(a_1,\ldots, a_m)\in \Bbb R^{md}$.