Projects per year
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 F_{σ }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.
Original language  English 

Pages (fromto)  49214936 
Number of pages  16 
Journal  Proceedings of the American Mathematical Society 
Volume  147 
Issue number  11 
Early online date  10 Jun 2019 
DOIs  
Publication status  Published  Nov 2019 
Keywords
 Weak tangent
 Microset
 Hausdorff dimension
 Assouad type dimensions
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Dive into the research topics of 'On the Hausdorff dimension of microsets'. Together they form a unique fingerprint.Projects
 2 Finished

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
1/02/18 → 11/06/21
Project: Standard

Fractal Geometry and Dimension: Fractal Geometry and dimension theory
1/09/16 → 30/06/18
Project: Fellowship