Real Analysis Exchange

Category of density points of fat Cantor sets.

Zoltán Buczolich

Full-text: Open access

Abstract

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 $D_{+}(P)=\cup_{\gamma>0}D_{\gamma}(P)$ is of first category in $Q.$

Article information

Source
Real Anal. Exchange, Volume 29, Number 1 (2003), 497-502.

Dates
First available in Project Euclid: 9 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149860211

Mathematical Reviews number (MathSciNet)
MR2063091

Zentralblatt MATH identifier
1063.28002

Subjects
Primary: 28A15: Abstract differentiation theory, differentiation of set functions [See also 26A24]
Secondary: 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 54E52: Baire category, Baire spaces

Keywords
Baire category density point nowhere dense perfect set Cantor set

Citation

Buczolich, Zoltán. Category of density points of fat Cantor sets. Real Anal. Exchange 29 (2003), no. 1, 497--502. https://projecteuclid.org/euclid.rae/1149860211


Export citation