The Falconer conjecture [F86] asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set ∆(E) has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean distance replaced by non-Euclidean norms ‖ · ‖X in which the unit ball is a polygon with 2K sides. We prove that for any such norm, and for any α ...

We consider the disintegration of the Lebesgue measure on the graph of a convex function f : R n → R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the k-dimensional Hausdorff measure of the k-dimensional face on which it is concentrated. The remarkable fact is that a prio...

Abstract We prove that the hitting measure is singular with respect to Lebesgue for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, Hausdorff dimension of strictly less than one. Equivalently, inequality between entropy and drift strict. A similar statement proven Coxeter