Assouad dimension and local structure of self-similar sets with overlaps in $\mathbb{R}^d$

For a self-similar set in $\mathbb{R}^d$ that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least $1$. In this paper, it is shown that the Assouad dimension of such a set is the sum of the dimension of the vector space spanned by the set of $\textit{overlapping directions}$ and the Assouad dimension of the orthogonal projection of the set self-similar set onto the orthogonal complement of that vector space. This result is applied to give sufficient conditions on the orthogonal parts of the similarities so that the self-similar set has Assouad dimension bigger than $2$, and also to answer a question posed by Farkas and Fraser. The result is also extended to the context of graph directed self-similar sets. The proof of the result relies on finding an appropriate weak tangent to the set. This tangent is used to describe partially the topological structure of self-similar sets which are both attractors of an iterated function system not satisfying the weak separation property and of an iterated functions system satisfying the open set condition.