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

Abstract

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

Cited by in corpus (1)

• A nonlinear version of the Newhouse thickness theorem