On arithmetic sums of fractal sets in R d

A compact set E ⊂ R dis said to be arithmetically thick if there exists a positive integer n so that the n ‐fold arithmetic sum of E has non‐empty interior. We prove the arithmetic thickness of E , if E is uniformly non‐flat, in the sense that there existsε 0 > 0 such that for x ∈ E and 0 < r ⩽ diam ( E ) , E ∩ B ( x , r ) never staysε 0 r ‐close to a hyperplane in R d . Moreover, we prove the arithmetic thickness for several classes of fractal sets, including self‐similar sets, self‐conformal sets in R d (with d ⩾ 2 ) and self‐affine sets in R 2 that do not lie in a hyperplane, and certain self‐affine sets in R d (with d ⩾ 3 ) under specific assumptions.