Unrectifiable 1-Sets with Moderate Essential Flatness Satisfy Besicovitch's 12-Conjecture

Abstract

AbstractIn this paper we show that for a wide class of totally unrectifiable 1-sets in the plane (or even a Hilbert space) satisfying a mild measure-theoretic flatness condition almost everywhere, at sufficiently small scales, the lower spherical density is bounded above by 12 at almost every point, thereby affirming Besicovitch's conjecture, which states that for all totally unrectifiable 1-sets in the plane (or possibly even in Rn, or a Hilbert space), the lower spherical density is bounded above by 12 at almost every point