2023/05/26 by Oona Rainio, Toshiyuki Sugawa, Rainio, Oona +3
Mathematics · #30F45 (Primary) 30C85 (Secondary) #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2305.16723
openalex publication_date 2023/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℝn 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 ℝn 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.