The Minkowski sum of linear Cantor sets
Volume 212 / 2024
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 uniqueness, that is, the set of $z \in [0,2]$ which have a unique representation as $z = x + y$ with $x, y \in C$. It is not difficult to show that $\dim _H (U) = \log(2)/\!\log (3)$ and $U$ essentially looks like $2C$.
Assume $A \subset \{0,1, \dots , n-1\}$ and further that $0, n-1 \in A$. Define the linear Cantor set $C_A = C_{A,n}$, that is, the set of numbers whose base $n$ representations contain only digits from $A$. In symbols, $C_{A,n} := \{\sum _{i = 1}^\infty {a_i}/{n^i}: a_i \in A\}$. 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 on the properties of the set of uniqueness $U_A$.
