Points de densité d'ensembles de cantor

Abstract

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 Δ_i1,…,in, i_j = 0, 1 after the nth step. At the (n + 1)th step we remove from each Δ_i1,…,in an interval J_i1,…,in of length 2⁻ⁿu_n > 0 located at the center of Δ_i1,…,in, with the assumption Σ_n=0^∞ u_n < 1 that guarantees |C| > 0. A point χ ϵ C is characterized by the sequence (i_n) such that χ ϵ Δ_i1,…,in for every n. Let m(x, n) be the largest integer m < n such that i_m+1 ≠ i_n. We prove that χ is a density point for C iff

$$\text{lim}\text{n→∞}\text{2}{}^{\mathrm{2-m(x,\; n)}}\text{· u}{}_{\mathrm{m(x,\; n)}}\text{= 0.}$$