The Minkowski sum of linear Cantor sets

Abstract

Let $C$ be the classical middle third Cantor set. It is well known that $C+C = [0,2]$ (Steinhaus, 1917). (Here $+$ denotes the Minkowski sum.) Let $U$ be the set of $z \in [0,2]$ which have a unique representation as $z = x + y$ with $x, y \in C$ (the set of uniqueness). It isn't difficult to show that $\dim_H U = \log(2) / \log(3)$ and $U$ essentially looks like $2C$. Assuming $0,n-1 \in A \subset \{0,1,\dots,n-1\}$, define $C_A = C_{A,n}$ as the linear Cantor set which the attractor of the iterated function system \[ \{ x \mapsto (x + a) / n: a \in A \}. \] We consider various properties of such linear Cantor sets. Our main focus will be on the structure of $C_{A,n}+C_{A,n}$ depending on $n$ and $A$ as well as the properties of the set of uniqueness $U_A$.