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 Fσ set ∆ ⊆ [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 ∆. Our work is motivated by the general programme of determining what geometric information about a set can be determined at the level of tangents.