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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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