Hausdorff dimension of the arithmetic sum of self-similar sets

TL;DR

This paper investigates the Hausdorff dimension of the sum of two self-similar sets generated by similitudes with a common base, showing it is either self-similar or an attractor of an infinite iterated function system, and provides exact dimension calculations.

Contribution

It characterizes the structure of the sum of two self-similar sets and derives conditions for exact Hausdorff dimension computation, extending previous results.

Findings

Sum set is either self-similar or an infinite IFS attractor.

Provides explicit Hausdorff dimension formulas under certain conditions.

Partially addresses dimension calculation without the irrational assumption.

Abstract

Let $\beta>1$. We define a class of similitudes \[S:=\left\{f_{i}(x)=\dfrac{x}{\beta^{n_i}}+a_i:n_i\in \mathbb{N}^{+}, a_i\in \mathbb{R}\right\}.\] Taking any finite similitudes $\{f_{i}(x)\}_{i=1}^{m} $ from $S$, it is well known that there is a unique self-similar set $K_1$ satisfying $K_1=\cup_{i=1}^{m} f_{i}(K_1)$. Similarly, another self-similar set $K_2$ can be generated via the finite contractive maps of $S$. We call $K_1+K_2=\{x+y:x\in K_1, y\in K_2\}$ the arithmetic sum of two self-similar sets. In this paper, we prove that $K_1+K_2$ is either a self-similar set or a unique attractor of some infinite iterated function system. Using this result we can then calculate the exact Hausdorff dimension of $K_1+K_2$ under some conditions, which partially provides the dimensional result of $K_1+K_2$ if the IFS's of $K_1$ and $K_2$ fail the irrational assumption, see Peres and Shmerkin \cite{PS}.