Real Analysis Exchange

Category of density points of fat Cantor sets.

Zoltán Buczolich

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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

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

First available in Project Euclid: 9 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Baire category density point nowhere dense perfect set Cantor set


Buczolich, Zoltán. Category of density points of fat Cantor sets. Real Anal. Exchange 29 (2003), no. 1, 497--502.

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