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".
Publié le : 2002-03-14
Classification:
Unrectifiable,
Besicovitch 1/2-problem,
rectifiable,
density,
quasi arc,
Hausdorff measure,
28
@article{1045578692,
author = {Farag, Hany M.},
title = {On the 1/2-Problem of Besicovitch:
quasi-arcs do not contain sharp saw-teeth},
journal = {Rev. Mat. Iberoamericana},
volume = {18},
number = {1},
year = {2002},
pages = { 17-40},
language = {en},
url = {http://dml.mathdoc.fr/item/1045578692}
}
Farag, Hany M. On the 1/2-Problem of Besicovitch:
quasi-arcs do not contain sharp saw-teeth. Rev. Mat. Iberoamericana, Tome 18 (2002) no. 1, pp. 17-40. http://gdmltest.u-ga.fr/item/1045578692/