On arithmetic sums of fractal sets in ${\Bbb R}^d$

Abstract

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.