Unrectifiable 1-sets with moderate essential flatness satisfy Besicovitch's \(\frac 12\)-conjecture
The paper under review deals with the long-standing Besicovitch's \(\frac 12\)-conjecture according to which the lower density is typically bounded above by \(\frac 12\) for all totally unrectifiable \(1\)-sets in the plane. A 1-set has positive and finite 1-dimensional Hausdorff measure, and a totally unrectifiable set meets every rectifiable curve in a set of 1-dimensional Hausdorff measure zero. Let \(\sigma_1(\mathbb R^2)\) be the smallest number such that if \(E\subset\mathbb R^2\) is a totally unrectifiable 1-set, then the lower density is bounded above by \(\sigma_1(\mathbb R^2)\) at almost every point in \(E\). In [Math. Ann. 115, 296-329 (1938; Zbl 0018.11302)] \textit{A. S. Besicovitch} proved that \(\frac 12\leq\sigma_1(\mathbb R^2)\leq\frac 34\) and conjectured that \(\sigma_1(\mathbb R^2)=\frac 12\). In a tricky proof the author verifies this conjecture for a subclass of totally unrectifiable \(1\)-sets consisting of sets which are flat almost everywhere. The notation of flatness used to characterize the subclass is related to the earlier work by \textit{P. W. Jones} [Invent. Math. 102, No. 1, 1-15 (1990; Zbl 0731.30018)].
scientific article; zbMATH DE number 1422991
scientific article