The Pointwise Densities of the Cantor Measure

Density theorems for Hausdorff and packing measures of self-similar sets

“…[3,6]. The local behaviour of the s-dimensional Hausdorff measure of K has also been investigated by various authors in [1,5,6,7,8,13,17]. For example, if we write H s for the s-dimensional Hausdorff measure, then Hutchinson in [6] proved that if the OSC is satisfied, then there exist positive constants c 1 , c 2 , r 0 such c 1 (2r) s ≤ H s (K ∩ B(x, r)) ≤ c 2 (2r) s for all x ∈ K and all 0 < r < r 0 .…”

On the pointwise densities of the Cantor measure

“…To our knowledge, the Cantor measures in [1,9] are the only Ahlfors s-regular measures such that s is not an integer and the exact values of Θ * s (μ, x) and Θ s * (μ, x) are determined for every point x. However, [1,9] both require that the length of the gap is not less than the length of the basic interval of same order, i.e., 1 − 2a a.…”

“…Feng, Hua and Wen [1] first obtained the exact values of Θ * s (μ, x) and Θ s * (μ, x) for every point x ∈ K (a) for a ∈ (0, 1 3 ].…”

“…Based on [1], Li and Yao [9] investigated the non-symmetrical case. In addition, McCoy [10] studied more general cases, but the results in [10] only considered the exact values of Θ * s (μ, x) and Θ s * (μ, x) for μ-a.e.…”

The upper densities of symmetric perfect sets

“…Let C be the classical middle-third Cantor set and let µ C be the restriction of the Hausdorff measure H s over the set C. Feng et al [3] determined the exact upper s-density of µ C , i.e., Θ * s (µ C , x) = 2/4 s for µ C almost all x ∈ C.…”