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Symmetric porosity of symmetric Cantor sets. (English) Zbl 0814.26003
The main result of this paper is the theorem:
Let $${\mathcal C}\{\alpha_ n\}$$ be the symmetric Cantor set determined by a sequence $$\{\alpha_ n\}$$. 1. If $$\limsup \alpha_ n> {1\over 2}$$ or $$\{\alpha_ n\}$$ is not weakly sparse, then $${\mathcal C}\{\alpha_ n\}$$ is symmetrically porous. 2. If $${\mathcal C}\{\alpha_ n\}$$ is symmetrically porous, then $$\limsup \alpha_ n> {1\over 2}$$ or $$\{\alpha_ n\}$$ is not sparse.
A sequence $$\{\alpha_ n\}\subset [0,1)$$ is called sparse (weakly sparse) if, for each sequence $$\{\alpha_{n_ k}\}$$ having a positive limit inferior, the sequence $$\{n_ k- n_{k-1}\}$$ diverges to $$\infty$$ (is unbounded).
Reviewer: R.Pawlak (Łódź)
##### MSC:
 26A03 Foundations: limits and generalizations, elementary topology of the line 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets
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##### References:
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