Favard-length decay exponent

Description of constant

Let $E\subset \mathbb{R}^2$ be a planar set. The Favard length of $E$ is defined by

where $\mathrm{Proj}$ is orthogonal projection to the horizontal axis and $R_\theta$ is rotation by angle $\theta$. [NPV2011-def-fav]

The Favard length has the following probabilistic interpretation: up to a constant factor, $\mathrm{Fav}(E)$ is the probability that a “Buffon’s needle” (a long line segment dropped at random) hits $E$. [NPV2011-buffon-interpretation]

Let $C_n$ be the $n$-th stage in the construction of the middle-half Cantor set, and let

Then $K_n$ is a union of $4^n$ axis-parallel squares of side length $4^{-n}$. [NPV2011-Kn-4n-squares]

A classical theorem of Besicovitch implies that $\mathrm{Fav}(K_n)\to 0$ as $n\to\infty$, and it remains open to determine the exact rate of decay. [NPV2011-besicovitch-open-decay]

We define the Favard-length decay exponent $\alpha_{\mathrm{Fav}}$ by

The best established range currently is

[NPV2011-thm1-powerlaw] [BV2010-thm1-lb-logn-over-n]

Known upper bounds

Known lower bounds

Additional comments and links

• 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

• [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.”

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.