Favard-length decay exponent ($C_{60}$)

• Prior explicit upper bound. Before the power-law bound, the only explicit upper bound recorded in this literature was of iterated-log type, $\exp(-c\,\log_* n)$, attributed to Peres–Solomyak. [NPV2011-exp-logstar]

• Non-optimality and a method barrier. Nazarov–Peres–Volberg remark that the exponent $1/6$ is not optimal, but that decay faster than $O(n^{-1/4})$ would require new ideas (relative to their methods). [NPV2011-remark-n-1-4]

• Wikipedia page on Favard length

References (from the repository page)

• [BV2010] Bateman, Michael; Volberg, Alexander. An estimate from below for the Buffon needle probability of the four-corner Cantor set. Mathematical Research Letters 17 (2010), no. 5, 959–967. DOI: https://doi.org/10.4310/MRL.2010.v17.n5.a12. arXiv PDF: https://arxiv.org/pdf/0807.2953. Google Scholar

• [BV2010-thm1-lb-logn-over-n] loc: arXiv v1 PDF p.3, Theorem 1 (display (1.2)) quote: “There exists $c>0$ such that $\mathrm{Fav}(K_n)\ge c\,\frac{\log n}{n}$ for all $n\in\mathbb{N}$.”

• [NPV2011] Nazarov, Fedor; Peres, Yuval; Volberg, Alexander. The power law for the Buffon needle probability of the four-corner Cantor set. St. Petersburg Mathematical Journal 22 (2011), no. 1, 61–72. DOI: https://doi.org/10.1090/S1061-0022-2010-01133-6. arXiv PDF: https://arxiv.org/pdf/0801.2942. Google Scholar

• [NPV2011-besicovitch-open-decay] loc: arXiv v1 PDF p.1, Abstract (sentences on Besicovitch and open problem) quote: “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.”

• [NPV2011-exp-logstar] loc: arXiv v1 PDF p.1, Abstract (sentence on iterated-log upper bound) quote: “Until recently, the only explicit upper bound was $\exp(-c\,\log_* n)$, due to Peres and Solomyak.”

• [NPV2011-def-fav] loc: arXiv v1 PDF p.2, equation (1.1) quote: “$\mathrm{Fav}(E)=\frac{1}{\pi}\int_0^\pi \lvert \mathrm{Proj}\,R_\theta E\rvert\,d\theta$.”

• [NPV2011-buffon-interpretation] loc: arXiv v1 PDF p.2, paragraph after (1.1) quote: “it is the probability that the “Buffon’s needle,” a long line segment dropped at random, hits $E$”

• [NPV2011-Kn-4n-squares] loc: arXiv v1 PDF p.2, paragraph after the Buffon-needle interpretation quote: “The set $K_n=C_n^2$ is a union of $4^n$ squares with side length $4^{-n}$.”

• [NPV2011-thm1-powerlaw] loc: arXiv v1 PDF p.3, Theorem 1 quote: “For every $\delta>0$, there exists $C>0$ such that $\mathrm{Fav}(K_n)\le C n^{\delta-1/6}$ for all $n\in\mathbb{N}$.”

• [NPV2011-remark-n-1-4] loc: arXiv v1 PDF p.3, Remarks (bullet after Theorem 1) quote: “a bound decaying faster than $O(n^{-1/4})$ would require new ideas.”

Repository table (commit 2c1968c, 26 September 2026): best known lower bound $\frac{1}{6}$; best known upper bound 1. Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds