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Uniformly perfect sets, Hausdorff dimension, and conformal capacity
- Publication Year :
- 2023
-
Abstract
- Using the definition of uniformly perfect sets in terms of convergent sequences, we apply lower bounds for the Hausdorff content of a uniformly perfect subset $E$ of $\mathbb{R}^n$ to prove new explicit lower bounds for the Hausdorff dimension of $E.$ These results also yield lower bounds for capacity test functions, which we introduce, and enable us to characterize domains of $\mathbb{R}^n$ with uniformly perfect boundaries. Moreover, we show that an alternative method to define capacity test functions can be based on the Whitney decomposition of the domain considered.<br />Comment: 29 pages, 2 figures
- Subjects :
- Mathematics - Complex Variables
30F45 (Primary) 30C85 (Secondary)
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2305.16723
- Document Type :
- Working Paper