Hausdorff measures for a class of homogeneous cantor sets

Exact Hausdorff and Packing measure of certain Cantor sets, not necessarily self-similar or homogeneous

“…If we assume this condition of decreasing gaps, then we can not have any gap with length zero, and then condition 2 is also satisfied. Therefore, the main result in [16] is a corollary of Theorem 1.…”

“…This estimate is based on the local behaviour of the measure µ expressed by a generalization of the mass distribution principle instead of the upper density (see Lemma 3 below). We follow the approach of [16,13].…”

“…However, some sets considered by Qu et al do not satisfy the hypothesis required in [13]. In the present article, we consider Cantor sets which might not be self-similar or homogeneous, including the considered in [16] and in [13] as well as examples not considered before and establish a formula for the exact Hausdorff measure (see Theorem 1). The hypothesis assumed are of separation type and decay on the lengths of gaps.…”

“…Ayer and Strichartz [1] computed the exact Hausdorff measure of self-similar Cantor sets satisfying weaker conditions than the open set condition and gave very precise algorithms to obtain it. Qu et al [16], have determined the Hausdorff measure of homogeneous (meaning that the gaps have all the same length) Cantor sets including the family considered by Marion and some non self-similar sets. Recently, Pedersen et al [13] have partially extended the result by Qu et al, computing the exact Hausdorff measure of a family of Cantor sets not previously considered, including non-homogeneous ones.…”