English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $\Cant_n$ be the $n$-th generation in the construction of the middle-halfCantor set. The Cartesian square $\K_n = \Cant_n \times \Cant_n$ consists of$4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown atrandom in the unit square will meet $\K_n$ is essentially the average length ofthe projections of $\K_n$, also known as the Favard length of $\K_n$. Aclassical theorem of Besicovitch implies that the Favard length of $\K_n$ tendsto 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 toPeres and Solomyak. ($\log_* n$ is the number of times one needs to take log toobtain a number less than 1 starting from $n$). In Nazarov-Peres-Volberg paper(arxiv:math 0801.2942) the power estimate from above was obtained. The exponentin this paper was less than 1/6 but could have been slightly improved. On theother hand, a simple estimate shows that from below we have the estimate$\frac{c}{n}$. Here we apply the idea from papers of Nets Katz (MRL (1996),527-536) and Bateman-Katz (arxiv:math/0609187v1 2006) to show that the estimatefrom below can be in fact improved to $c \frac{\log n}{n}$. This is in drasticdifference from the case of {\em random} Cantor sets studied by Peres andSolomyak in Pacific J. Math. 204 (2002), 473-496.

An estimate from below for the Buffon needle probability of the four-corner Cantor set