Category of density points of fat Cantor sets. (English) Zbl 1063.28002

Buczolich, Zoltán

Real Anal. Exch. 29(2003-2004), No. 1, 497-502 (2004).

Summary: Denote by \(D_\gamma(P)\) the set of those points where the lower Lebesgue density of \(P\subset\mathbb{R}\) is bigger or equal than \(\gamma\). We show that if \(\gamma> 0.5\) then \(D_\gamma(P)\cap P\) is always of first category in any nowhere dense perfect set \(P\). On the other hand, there exists a fat Cantor set \(Q\) which is a subset of \(D_{0.5}(Q)\) while for other fat Cantor sets \(P\) it is possible that \(D_+(P)= \bigcup_{\gamma> 0} D_\gamma(P)\) is of first category in \(Q\).

MSC:

28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A75	Length, area, volume, other geometric measure theory
54E52	Baire category, Baire spaces

Keywords:

Baire category; density point; Cantor set; nowhere dense perfect set