Typical self-affine sets with non-empty interior

Abstract

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