The Minkowski sum of linear Cantor sets

TL;DR

This paper investigates the Minkowski sums of linear Cantor sets generated by specific iterated function systems, analyzing their structure and the properties of the set of points with unique representations.

Contribution

It introduces a comprehensive study of Minkowski sums of linear Cantor sets and characterizes the structure of the set of unique representations based on the parameters.

Findings

The structure of $C_{A,n}+C_{A,n}$ varies with $n$ and $A$.

The set of unique representations $U_A$ exhibits specific fractal properties.

Conditions under which the sum set is an interval or fractal are identified.

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 $U_A$.