On the Assouad dimension of differences of self-similar fractals

TL;DR

This paper investigates the Assouad dimension of difference sets of self-similar fractals on the real line, showing that under certain conditions, the dimension of the difference set is at most twice that of the original set.

Contribution

It establishes a bound on the Assouad dimension of difference sets for self-similar sets satisfying a weak separation condition, extending understanding of fractal difference sets.

Findings

Assouad dimension of differences is bounded by twice the original under certain conditions.

The result applies to a class of asymmetric Cantor sets.

Provides new bounds for fractal difference sets.

Abstract

If $X$ is a set with finite Assouad dimension, it is known that the Assouad dimension of $X-X$ does not necessarily obey any non-trivial bound in terms of the Assouad dimension of $X$. In this paper, we consider self-similar sets on the real line and we show that if a particular weak separation condition is satisfied, then the Assouad dimension of the set of differences is bounded above by twice the Assouad dimension of the set itself. We then apply this result to a particular class of asymmetric Cantor sets.