  A compact set $E\subset {\Bbb R}^d$ 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 $\epsilon_0>0$ such that for
$x\in E$ and $0<r\leq {\rm diam}(E)$, $E\cap B(x,r)$ never stays
$\epsilon_0r$-close to a hyperplane in ${\Bbb R}^d$. Moreover, we prove the
arithmetic thickness for several classes of fractal sets, including
self-similar sets, self-conformal sets in ${\Bbb R}^d$ (with $d\geq 2$) and
self-affine sets in ${\Bbb R}^2$ that do not lie in a hyperplane, and certain
self-affine sets in ${\Bbb R}^d$ (with $d\geq 3$) under specific assumptions.
