Properties of Quasi-Assouad dimension

Abstract

The connections between quasi-Assouad dimension and tangents are studied. We apply these results to the calculation of the quasi-Assouad dimension for a class of planar self-affine sets. We also show that sets with decreasing gaps have quasi-Assouad dimension $0$ or $1$ and 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. Moreover, for closed sets, we show that the Hausdorff dimension is an upper bound for the lower-Assouad dimension.

Theorem 1 and its consequences were removed because it proof was not correct. Added a Proposition showing that for closed sets, the Hausdorff dimension is an upper bound for the lower-quasi Assouad dimension