Fractal sumset properties

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 \(K_1,K_2\),..., \(K_\ell\) 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 \(\mathbb{R}^d\).