  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 $\mathbb{R}^d$.
