On the Assouad dimension of differences of self-similar fractals

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.