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

Abstract In 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 1 2 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 R n , or a Hilbert space), the lower spherical density is bounded above by 1 2 at almost every point.