Quasisymmetric minimality on packing dimension for Moran sets

Packing Dimension, Hausdorff Dimension and Cartesian Product Sets

We show that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor (1994) is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor. AMS Classification numbers: Primary 28A78, 28A80.

Porous Sets and Quasisymmetric Maps

A set A in R" is called porous if there is a > 0 such that every ball B(x,r) contains a point whose distance from A is at least ar. We show that porosity is preserved by quasisymmetric maps, in particular, by bilipschitz maps. Local versions are also given.

Billingsley ’ S Packing Dimension

For a stochastic process on a finite state space, we define the notion of a packing measure based on the naturally defined cylinder sets. For any two measures ν, γ, corresponding to the same stochastic process, if F ⊆ { ω ∈ Ω : lim n log γ(cn(ω)) log ν(cn(ω)) = θ } , then we prove that Dimν(F ) = θ Dimγ(F ).

Conformal Dimension and the Quasisymmetric Geometry of Metric Spaces

Asymptotics of Best-packing on Rectifiable Sets

We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz s-energy configurations and determine the s-th root asymptotic behavi...