Comparing Packing Measures to Hausdorff Measures on the Line

Abstract

For each 0 < s < 1, define where ${\cal P}^s$, ${\cal H}^s$ denote respectively the s‐dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E ⊂ R with $0 < {\cal H}^s (E) < \infty$. In this paper we give a nontrivial estimation of c(s), namely, $2^s (1+v(s))^s \le c(s) \le 2^s \left (2 ^{1 \over s} - 1 \right )$ for each 0 < s < 1, where $v(s) = \min \left \{ 16 ^{- {{1} \over {1-s}}}, 8 ^{- {{1} \over {(1-s)^2}}} \right \}$. As an application, we obtain a lower density theorem for Hausdorff measures.

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

“…Introduced by Tricot [33], the concept of Packing measure and dimension for fractals was studied by several authors (see e.g. [2,[6][7][8] and references therein). Let ε > 0 and I ⊂ N. An ε-packing of F is a collection of disjoint balls (B i ) i∈I with diameter at most ε and midpoints of B i placed in F .…”

“…Upper and lower bounds for M were given by Feng[6]. He also raised the unsolved question if M = 2 D (2 − 1) D is true or not.Remark 3.2.It remains an open question, if for homogeneous one-dimensional Cantor sets and their related uniform distributions in the non-dyadic case (i.e.…”

On Cantor Sets and Packing Measures

“…On the other hand, for the packing measure P n and the Hausdorff measure H n on R n we have Ω n H n = Ω n P n = L n , where L n is the Lebesgue measure on R n and Ω n = L n (B(0, 1/2)). For further study on the relationship between packing measure and Hausdorff measure we refer to Feng [4] and Rajala [12].…”