In this talk we introduce two notions of fractal sumset properties. A compact set K in R^d is said to have the Hausdorff sumset property (HSP) if for any n there exist sets K_1, K_2,…, K_n such that K_1+K_2+…+K_n\subset K and each K_i has the same Hausdorff dimension as K. Analogously, we say a compact set K to have the packing sumset property (PSP) if we replace the Hausdorff dimension by the packing dimension in the definition of HSP. 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. This is joint work with Zhiqiang Wang.