Dans Bateman et Volberg (2008) [1], on a démontré que la longueur de Favard de la stage n-ième d'ensemble 1/4 de Cantor décroit au plus comme . Dans Bond et Volberg (2008) [2], on a introduit une longueur circulaire de Favard, et on a démontré que les même estimations sont valable, au moins si le rayon du cercle satisfait . Le résulat de Bond et Volberg (2008) [2] mene naturallement à une hypothèse qui (si soit valable) donne la preuve que le résultat concernant la fonction maximale circulaire de Seeger, Tao et Wright (2005) [3] est exact.
In Bateman and Volberg (2008) [1], it was shown that the n-th partial 1/4 Cantor in the plane set decays in Favard length no faster than . In Bond and Volberg (2008) [2], the so-called circular Favard length of the same set is studied, and the same estimate is shown to persist when the circle has radius . By considering characteristic functions, the result of Bond and Volberg (2008) [2] naturally leads to a conjecture which (if true) would imply the sharpness of the boundedness of the circular maximal operator proved by Seeger, Tao and Wright (2005) [3].
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@article{CRMATH_2010__348_17-18_963_0, author = {Bond, Matthew}, title = {How likely is {Buffon's} ring toss to intersect a planar {Cantor} set?}, journal = {Comptes Rendus. Math\'ematique}, pages = {963--966}, publisher = {Elsevier}, volume = {348}, number = {17-18}, year = {2010}, doi = {10.1016/j.crma.2010.08.002}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2010.08.002/} }
TY - JOUR AU - Bond, Matthew TI - How likely is Buffon's ring toss to intersect a planar Cantor set? JO - Comptes Rendus. Mathématique PY - 2010 SP - 963 EP - 966 VL - 348 IS - 17-18 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.08.002/ DO - 10.1016/j.crma.2010.08.002 LA - en ID - CRMATH_2010__348_17-18_963_0 ER -
%0 Journal Article %A Bond, Matthew %T How likely is Buffon's ring toss to intersect a planar Cantor set? %J Comptes Rendus. Mathématique %D 2010 %P 963-966 %V 348 %N 17-18 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.08.002/ %R 10.1016/j.crma.2010.08.002 %G en %F CRMATH_2010__348_17-18_963_0
Bond, Matthew. How likely is Buffon's ring toss to intersect a planar Cantor set?. Comptes Rendus. Mathématique, Tome 348 (2010) no. 17-18, pp. 963-966. doi : 10.1016/j.crma.2010.08.002. http://www.numdam.org/articles/10.1016/j.crma.2010.08.002/
[1] An estimate from below for the Buffon needle probability of the four-corner Cantor set, 2008 (pp. 1–11) | arXiv
[2] Estimates from below of the Buffon noodle probability for undercooked noodles, 2008 (pp. 1–10) | arXiv
[3] A. Seeger, T. Tao, J. Wright, Notes on the lacunary spherical maximal function, preprint, 2005, pp. 1–14.
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