On the 1/2-Problem of Besicovitch: quasi-arcs do not contain sharp saw-teeth

Hany Farag
2002 Revista matemática iberoamericana  
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
more » ... 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".
doi:10.4171/rmi/310 fatcat:khj247yftvfc3p435ypqxru374