The power law for the Buffon needle probability of the four-corner Cantor set
| dc.creator | Nazarov, Fedor | |
| dc.creator | Peres, Yuval | |
| dc.creator | Volberg, Alexander | |
| dc.date | 2008-01-18 | |
| dc.date.accessioned | 2026-07-07T08:55:22Z | |
| dc.date.available | 2026-07-07T08:55:22Z | |
| dc.description | Let $C_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $K_n$ of $C_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $K_n$ is essentially the average length of the projections of $K_n$, also known as the Favard length of $K_n$. A classical theorem of Besicovitch implies that the Favard length of $K_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\exp(- c\log_* n)$, due to Peres and Solomyak. ($\log_* n$ is the number of times one needs to take log to obtain a number less than 1 starting from $n$). We obtain a power law bound by combining analytic and combinatorial ideas. | |
| dc.description | 16 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/0801.2942 | |
| dc.identifier | http://arxiv.org/abs/0801.2942 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/146250 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Metric Geometry | |
| dc.subject | 28A80; 28A75 | |
| dc.title | The power law for the Buffon needle probability of the four-corner Cantor set | |
| dc.type | text |