Points de densité dʹensembles de cantor

In this note we give a combinatorial characterization of the set of density points for a class of triadic Cantor sets C with positive measure. As is usual in triadic constructions, there are 2n remaining intervals Δi1,…,in, ij = 0, 1 after the nth step. At the (n + 1)th step we remove from each Δi1,…,in an interval Ji1,…,in of length 2−nun > 0 located at the center of Δi1,…,in, with the assumption Σn=0∞ un < 1 that guarantees |C| > 0. A point χ ϵ C is characterized by the sequence (in) such that χ ϵ Δi1,…,in for every n. Let m(x, n) be the largest integer m < n such that im+1 ≠ in. We prove that χ is a density point for C iff limn→∞ 22−m(x, n) · um(x, n) = 0.