Comptes Rendus
no. 17-18
Mathematical Analysis
How likely is Buffon's ring toss to intersect a planar Cantor set?
Gilles Pisier
Matthew Bond1
1Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 963-966

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 Clognn. 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 r⩾Cn. 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 LloglogL boundedness of the circular maximal operator proved by Seeger, Tao and Wright (2005) [3].

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 Clognn. 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 r⩾Cn. 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2010.08.002

Matthew Bond  1

1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 
Matthew Bond. How likely is Buffon's ring toss to intersect a planar Cantor set?. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 963-966. doi: 10.1016/j.crma.2010.08.002
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[1] M. Bateman; A. Volberg An estimate from below for the Buffon needle probability of the four-corner Cantor set, 2008 (pp. 1–11) | arXiv

[2] M. Bond; A. Volberg 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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