Fractal Sumset Properties

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 $\mathrm{\ell }\in {\mathbb{N}}_{\ge 2}$ there exist compact sets $K_1,K_2,\dots ,K_{\mathrm{\ell }}$ such that $K_1+K_2+\cdots +K_{\mathrm{\ell }}\subset K$ and ${\dim }_H{K}_i={\dim }_{H}K$ for all $1\le i\le \mathrm{\ell }$. 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 ${\mathbb{R}}^d$.