The box dimensions of exceptional self-affine sets in $\mathbb{R}^3$

TL;DR

This paper investigates the box dimensions of self-affine sets in three-dimensional space generated by permutation matrices, providing bounds and sharp results despite complex overlaps and non-sub-multiplicative functions.

Contribution

It introduces new techniques to analyze 3D self-affine sets, extending planar theories and addressing challenges like overlaps and singular value function issues.

Findings

Derived bounds for box dimensions with minimal assumptions

Achieved sharp results in many cases

Identified key challenges in extending planar theory to 3D

Abstract

We study the box dimensions of self-affine sets in $\mathbb{R}^3$ which are generated by a finite 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 $\mathbb{R}^3$ 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.