In this paper we give an alternative proof of our recent result that totally unrectifiable 1-sets which satisfy a measure-theoretic flatness condition at almost every point and sufficiently small scales, satisfy Besicovitch's 1/2-Conjecture which states that the lower spherical density for totally unrectifiable 1-sets should be bounded above by 1/2 at almost every point. This is in contrast to rectifiable 1-sets which actually possess a density equal to unity at almost every point. Our present method is simpler and is of independent interest since it mainly relies on general properties of finite sets of points satisfying a scale-invariant flatness condition. For instance it shows that a quasi-arc of small constant cannot contain "sharp saw-teeth".
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Hany M. Farag, On the 1/2-Problem of Besicovitch: quasi-arcs do not contain sharp saw-teeth. Rev. Mat. Iberoam. 18 (2002), no. 1, pp. 17–40DOI 10.4171/RMI/310