Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 655-673
University of Jyväskylä,
Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä,
Finland; tuomo.j.ojala 'at' jyu.fi
Abstract. We define (α)n-regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the necessary and sufficient conditions to be fat (or thin). In addition we discuss restrictions of doubling measures to these sets, and, in particular, give a sufficient condition to retain at least some of the restricted measures doubling on the set. Our main result generalizes and extends analogous results that were previously known to hold on the real line.
2010 Mathematics Subject Classification: Primary 28A12; Secondary 30L10.
Key words: Doubling measure, quasisymmetric map, thin set, fat set, Cantor set, dyadic cube.
Reference to this article: T. Ojala: On (α)n-regular sets. Ann. Acad. Sci. Fenn. Math. 39 (2014), 655-673.
doi:10.5186/aasfm.2014.3926
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