D. Gatzouras

Lacunarity of self-similar and stochastically self-similar sets

Let K be a self-similar set in Rdd, of Hausdorff dimension D, and denote by IK(c)l the d-dimensional Lebesgue measure of its 6-neighborhood. We study the limiting behavior of the quantity e-(d-D) IK(e)I as 6 -e 0. It turns out that this quantity does not have a limit in many interesting cases, including the usual ternary Cantor set and the Sierpinski carpet. We also study the above asymptotics for stochastically self-similar sets. The latter results then apply to zero-sets of stable bridges, which are stochastically self-similar (in the sense of the present paper), and then, more generally, to level-sets of stable processes. Specifically, it follows that, if Kt is the zero-set of a realvalued stable process of index a C (1,2], run up to time t, then e -/ SIIt(e)I converges to a constant multiple of the local time at 0, simultaneously for all t > 0, on a set of probability one. The asymptotics for deterministic sets are obtained via the renewal theorem. The renewal theorem also yields asymptotics for the mean 1E[IK(c) ] in the random case, while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks.

On the lattice case of an almost-sure renewal theorem for branching random walks

We formulate and verify an almost-sure lattice renewal theorem for branching random walks, whose non-lattice analogue is originally due to Nerman. We also identify the limit in these renewal theorems (both lattice and non-lattice) as the limit of Kingmans well-known martingale multiplied by a deterministic factor.

Lower Bound for the Maximal Number of Facets of a 0/1 Polytope

AbstractLet

On the Maximal Number of Facets of 0/1 Polytopes

f_{n-1}(P)

A Large Deviations Approach to the Geometry of Random Polytopes

denote the number of facets of a polytope

On mixing and ergodicity in locally compact motion groups

P

On images of Borel measures under Borel mappings

in

Hausdorff and box dimensions of certain self-affine fractals

{\Bbb R}^n

Threshold for the volume spanned by random points with independent coordinates

. We show that there exist 0/1 polytopes

A spectral radius formula for the Fourier transform on compact groups and applications to random walks

P