Multidimensional self-affine sets: non-empty interior and the set of uniqueness

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Let M be a d× d contracting matrix. In this paper we consider the self-affine iterated function system {Mv-u, Mv+u}, where u is a cyclic vector. Our main result is as follows: if | M|≥ 2⁻¹/ᵈ, then the attractor Aₘ has non-empty interior. We also consider the set Uₘ of points in Aₘ which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of Uₘ is positive. For this special class the full description of Uₘ is given as well. This paper continues our work begun in two previous papers.