Density bounds for outer parallel domains of unit ball packings

We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of space covered by them.Comment: 22 pages, 1 figur

Skinning maps

Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally geodesic boundary, answering a question of Y. Minsky. This is proven via a filling theorem, which states that as one performs higher and higher Dehn fillings, the skinning maps converge uniformly on all of Teichmuller space. We also exhibit manifolds with totally geodesic boundaries whose skinning maps have diameter tending to infinity, as well as manifolds whose skinning maps have diameter tending to zero (the latter are due to K. Bromberg and the author). In the final section, we give a proof of Thurston's Bounded Image Theorem.Comment: 50 pages, 4 figures. v3. Major revision incorporating referees' comments. To appear in the Duke Mathematical Journal. v2. Cosmetic changes, minor corrections, inclusion of theorem with K. Bromber

Hitting probabilities of random covering sets in tori and metric spaces

We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we consider covering sets generated by balls and, in the torus, we deal with general analytic generating sets

On densities of lattice arrangements intersecting every i-dimensional affine subspace

In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrangement of its polar body intersecting every affine hyperplane. Consequently, he formulated a conjecture that can be seen as a dual analog of Minkowski's fundamental theorem, and which is strongly linked to the well-known Mahler-conjecture. Based on the covering minima of Kannan & Lov\'asz and a problem posed by Fejes T\'oth, we arrange Makai Jr.'s conjecture into a wider context and investigate densities of lattice arrangements of convex bodies intersecting every i-dimensional affine subspace. Then it becomes natural also to formulate and study a dual analog to Minkowski's second fundamental theorem. As our main results, we derive meaningful asymptotic lower bounds for the densities of such arrangements, and furthermore, we solve the problems exactly for the special, yet important, class of unconditional convex bodies.Comment: 19 page

Universal coding for transmission of private information

We consider the scenario in which Alice transmits private classical messages to Bob via a classical-quantum channel, part of whose output is intercepted by an eavesdropper, Eve. We prove the existence of a universal coding scheme under which Alice's messages can be inferred correctly by Bob, and yet Eve learns nothing about them. The code is universal in the sense that it does not depend on specific knowledge of the channel. Prior knowledge of the probability distribution on the input alphabet of the channel, and bounds on the corresponding Holevo quantities of the output ensembles at Bob's and Eve's end suffice.Comment: 31 pages, no figures. Published versio