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

TL;DR

This paper establishes conditions under which certain fractal sets in Euclidean spaces become arithmetically thick, meaning their finite sums have non-empty interior, by analyzing geometric properties like non-flatness and applying to various fractal classes.

Contribution

It proves that uniformly non-flat fractal sets in ${ m I ext{-}R}^d$ are arithmetically thick, extending results to self-similar, self-conformal, and certain self-affine fractals.

Findings

Uniform non-flatness implies arithmetic thickness.

Self-similar and self-conformal sets are arithmetically thick.

Certain self-affine sets in ${ m I ext{-}R}^d$ are arithmetically thick under conditions.

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.