Hausdorff Dimension and Gaussian Fields

Hitting Probabilities and the Hausdorff Dimension of the Inverse Images of Anisotropic Gaussian Random Fields

Let X = {X(t), t ∈ RN} be a Gaussian random field with values in R defined by X(t) = ( X1(t), . . . , Xd(t) ) , where X1, . . . , Xd are independent copies of a centered Gaussian random field X0. Under certain general conditions on X0, we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse image X−1(F ), where F ⊆ R is a non-random Borel set. The class of G...

Historic set carries full hausdorff dimension

‎We prove that the historic set for ratio‎ ‎of Birkhoff average is either empty or full of Hausdorff dimension in a class of one dimensional‎ ‎non-uniformly hyperbolic dynamical systems.

Hausdorff Measure of the Sample Paths of Gaussian Random Fields

Hausdorff dimension and oracle constructions

Bennett and Gill (1981) proved that P 6= NP relative to a random oracle A, or in other words, that the set O[P=NP] = {A | P = NP} has Lebesgue measure 0. In contrast, we show that O[P=NP] has Hausdorff dimension 1. This follows from a much more general theorem: if there is a relativizable and paddable oracle construction for a complexity-theoretic statement Φ, then the set of oracles relative t...

Extremal Manifolds and Hausdorff Dimension

The best possible lower bound for the Hausdorff dimension of the set of v-approximable points on a C extremal manifold is obtained.