On the Hausdorff dimension of microsets

Abstract

We investigate how the Hausdorff dimensions of microsets are related to the dimensions of the original set. It is known that the maximal dimension of a microset is the Assouad dimension of the set. We prove that the lower dimension can analogously be obtained as the minimal dimension of a microset. In particular, the maximum and minimum exist. We also show that for an arbitrary ${\mathcal{F}}_{\sigma }$ set $\Delta \subseteq [0,d]$ containing its infimum and supremum there is a compact set in $[0,1{]}^d$ for which the set of Hausdorff dimensions attained by its microsets is exactly equal to the set $\Delta$. Our work is motivated by the general programme of determining what geometric information about a set can be determined at the level of tangents.

arXiv comment15 pages, 1 figure. small fixes

References in corpus (9)

• Local entropy averages and projections of fractal measures
• Assouad dimension of self-affine carpets
• Weak separation condition, Assouad dimension, and Furstenberg homogeneity
• Dynamics on fractals and fractal distributions
• Rigidity of quasisymmetric mappings on self-affine carpets
• Arithmetic patches, weak tangents, and dimension
• Dynamics of the scenery flow and geometry of measures
• Assouad type dimensions and homogeneity of fractals
• Locally rich compact sets

Cited by in corpus (4)

• Slices of the Takagi function
• Finer geometry of planar self-affine sets
• Regularity dimensions: quantifying doubling and uniform perfectness
• The range of dimensions of microsets