Category of density points of fat Cantor sets.

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 $Q.$