A GENERALISED CONICAL DENSITY THEOREM FOR UNRECTIFIABLE SETS

Abstract

We provide a strengthening of an elementary technique in geometric measure theory. Given an s -set S‰ R n , in the language of tangent measures, this technique establishes the existence of tangent measures to the measure H s bS on one side of an (ni 1) -plane. If S is purely unrectiflable or of dimension less than ni 1 , our strengthening consists of being able to flnd tangent measures on one side of an (ni 1) -plane containing a vector of our choice. We give an application of this result to the problem of characterising removable sets for harmonic functions.