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.

我们研究了自仿射集的盒子尺寸 ${\mathbb{[R}}^3$它们是由广义置换矩阵的有限集合生成的。我们以非常小的假设获得了维数的界限，并在许多情况下产生了清晰的结果。将完善的平面理论扩展到${\mathbb{[R}}^3$包括主要的平面投影是具有重叠的自仿射集（而不是自相似集）的仿射变形，并且自然修饰的奇异值函数通常不能进行次乘法。我们介绍了一些新技术来处理这些问题，并希望对进一步扩展该理论所面临的挑战提供一些见识。