Typical self-affine sets with non-empty interior

Abstract

Let $T_1, \dotsc , T_m$ be a family of $d \times d$ invertible real matrices with $\rVert T_i \rvert \lt 1/2$ 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 $\lbrace T_i x + a_i \rbrace$ on $\mathbb{R}^d$ has non-empty interior for almost all $(a_1 , \dotsc , a_m) \in \mathbb{R}^{md}$.