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Properties of Quasi-Assouad dimension (1703.02526v3)
Published 7 Mar 2017 in math.CA
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.