The box dimensions of exceptional self-affine sets in ℝ3

Abstract

We study the box dimensions of self-affine sets in ℝ3 which are generated by afinite collection of generalised permutation matrices. We obtain bounds for the dimensions which hold with very minimal assumptions and give rise to sharp results in many cases. There are many issues in extending the well-established planar theory to R3 including that the principal planar projections are (affine distortions of) self-affine sets with overlaps (rather than self-similar sets) and that the natural modified singular value function fails to be sub-multiplicative in general. We introduce several new techniques to deal with these issues and hopefully provide some insight into the challenges in extending the theory further.