Assouad dimension influences the box and packing dimensions of orthogonal projections

Abstract

We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of $F \subseteq \rn$ is no greater than $m$, then the box and packing dimensions of $F$ are preserved under orthogonal projections onto almost all $m$-dimensional subspaces. We also show that the threshold $m$ for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian.

10 pages, 1 figure. To appear in Journal of Fractal Geometry

References in corpus (1)

• A capacity approach to box and packing dimensions of projections of sets and exceptional directions

Cited by in corpus (2)

• Assouad type dimensions of infinitely generated self-conformal sets
• Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line