Properties of Quasi-Assouad dimension

Ignacio García, Kathryn Hare

Published 2017-03-07Version 1

It is shown that for controlled Moran constructions in $\mathbb{R}$, including the (sub) self-similar and more generally, (sub) self-conformal sets, the quasi-Assouad dimension coincides with the upper box dimension. This can be extended to some special classes of self-similar sets in higher dimensions. The connections between quasi-Assouad dimension and tangents are studied. We show that sets with decreasing gaps have quasi-Assouad dimension $0$ or $1$ and we exhibit an example of a set in the plane whose quasi-Assouad dimension is smaller than that of its projection onto the $x$-axis, showing that quasi-Assouad dimension may increase under Lipschitz mappings.