Fractal sumset properties

We introduce two notions of fractal sumset properties.A compact set \(K\subset\mathbb{R}^d\) $K\subset {\mathbb{R}}^d$ is said to have the Hausdorff sumset property (HSP) if for any \(\ell\in\mathbb{N}_{\ge 2}\) $\ell \in {\mathbb{N}}_{\ge 2}$ there exist compact sets \(K_1,K_2\) $K_1,K_2$ ,..., \(K_\ell\) $K_{\ell }$ such that \(K_1+K_2+\cdots+K_\ell\subset K\) $K_1+K_2+\cdots +K_{\ell }\subset K$ and \(\dim_H K_i=\dim_H K\) ${dim}_H{K}_i={dim}_{H}K$ for all \(1\le i\le \ell\) $1\le i\le \ell$ .Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set \(K\subset\mathbb{R}^d\) $K\subset {\mathbb{R}}^d$ 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 \(\mathbb{R}^d\) ${\mathbb{R}}^d$ .