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A compact set E ⊂ R d is 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 exists 0 > 0 such that for x ∈ E and 0 < r diam(E), E ∩ B(x, r) never stays 0r-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.