On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the \emph{weak separation property} is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the \emph{weak separation property} is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the \emph{weak separation property} is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a \emph{weak tangent} that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the `equality/maximal' dichotomy does not extend to this setting.