Fractal Sumset Properties

TL;DR

This paper introduces and compares two fractal sumset properties, showing that the Hausdorff sumset property fails for certain self-similar sets, whereas the packing sumset property holds broadly.

Contribution

It defines the Hausdorff and packing sumset properties for fractal sets and establishes the failure of HSP and the validity of PSP for specific classes of self-similar sets.

Findings

HSP fails for certain homogeneous self-similar sets with strong separation.

PSP holds for all homogeneous self-similar sets in .

Differentiates between behaviors of Hausdorff and packing sumset properties.

Abstract

In this paper we introduce two notions of fractal sumset properties. A compact set $K\subset\mathbb{R}^d$ is said to have the Hausdorff sumset property (HSP) if for any $\ell\in\mathbb{N}_{\ge 2}$ there exist compact sets $K_1, K_2,\ldots, K_\ell$ such that $K_1+K_2+\cdots+K_\ell\subset K$ and $\dim_H K_i=\dim_H K$ for all $1\le i\le \ell$. Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set $K\subset\mathbb{R}^d$ is said to have the packing sumset property (PSP). We show that the HSP fails for certain homogeneous self-similar sets satisfying the strong separation condition, while the PSP holds for all homogeneous self-similar sets in $\mathbb{R}^d$.