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 $U_A$.

Added some additional relevant references. Emphasized that almost all z \in C_A + C_A have a continuum of representations as z = x + y with x, y \in C_A. Added the observation that dim_H(U_A) < 1 for trivial reasons