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arXiv · 1911.04857

Assouad dimension influences the box and packing dimensions of orthogonal projections

Kenneth J. Falconer·Jonathan M. Fraser·Pablo Shmerkin
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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.

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Kenneth J. Falconer, Jonathan M. Fraser, Pablo Shmerkin. 2019-11-12. Assouad dimension influences the box and packing dimensions of orthogonal projections. https://arxiv.org/abs/1911.04857

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