Fractal Sumset Properties

In this paper we introduce two notions of fractal sumset properties. A compact set K⊂ℝ^d is said to have the Hausdorff sumset property (HSP) if for any ℓ∈ℕ_≥ 2 there exist compact sets K_1, K_2,…, K_ℓ such that K_1+K_2+⋯+K_ℓ⊂ K and _H K_i=_H K for all 1≤ i≤ℓ. Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set K⊂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 ℝ^d.