Regularity of non-autonomous self-similar sets

Non-autonomous self-similar sets are a family of compact sets which are, in some sense, highly homogeneous in space but highly inhomogeneous in scale. The main purpose of this note is to clarify various regularity properties and separation conditions relevant for the fine local scaling properties of these sets. A simple application of our results is a precise formula for the Assouad dimension of non-autonomous self-similar sets in ${\mathbb{R}}^d$ satisfying a certain ``bounded neighbourhood’’ condition, which generalizes earlier work of Li–Li–Miao–Xi and Olson–Robinson–Sharples. We also see that the bounded neighbourhood assumption is, in few different senses, as general as possible.