Favard-length decay exponent

Description of constant

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

$Fav (E) := \frac{1}{π} \int_{0}^{π} ∣ Proj R_{θ} E ∣ d θ,$

where $Proj$ is orthogonal projection to the horizontal axis and $R_{θ}$ is rotation by angle $θ$ . <a href="#NPV2011-def-fav">[NPV2011-def-fav]</a>

The Favard length has the following probabilistic interpretation: up to a constant factor, $Fav (E)$ is the probability that a “Buffon’s needle” (a long line segment dropped at random) hits $E$ . <a href="#NPV2011-buffon-interpretation">[NPV2011-buffon-interpretation]</a>

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

$K_{n} := C_{n} \times C_{n} .$

Then $K_{n}$ is a union of $4^{n}$ axis-parallel squares of side length $4^{- n}$ . <a href="#NPV2011-Kn-4n-squares">[NPV2011-Kn-4n-squares]</a>

A classical theorem of Besicovitch implies that $Fav (K_{n}) \to 0$ as $n \to \infty$ , and it remains open to determine the exact rate of decay. <a href="#NPV2011-besicovitch-open-decay">[NPV2011-besicovitch-open-decay]</a>

We define the Favard-length decay exponent $α_{Fav}$ by

$C_{60} := α_{Fav} := sup {α \geq 0 : \exists C > 0 such that Fav (K_{n}) \leq C n^{- α} for all n \in N} .$

The best established range currently is

$\frac{1}{6} \leq α_{Fav} \leq 1.$

<a href="#NPV2011-thm1-powerlaw">[NPV2011-thm1-powerlaw]</a> <a href="#BV2010-thm1-lb-logn-over-n">[BV2010-thm1-lb-logn-over-n]</a>

Known upper bounds

Bound	Reference	Comments
$1$	<a href="#BV2010">[BV2010]</a>	Bateman–Volberg prove $Fav (K_{n}) \geq c \frac{l o g n}{n}$ , which rules out any estimate $Fav (K_{n}) \leq C n^{- α}$ with $α > 1$ . <a href="#BV2010-thm1-lb-logn-over-n">[BV2010-thm1-lb-logn-over-n]</a>

Known lower bounds

Bound	Reference	Comments
$0$		Trivial (since $Fav (K_{n})$ is uniformly bounded).
$1/6$	<a href="#NPV2011">[NPV2011]</a>	Nazarov–Peres–Volberg prove $Fav (K_{n}) \leq C n^{- 1/6 + δ}$ (equivalently, $Fav (K_{n}) \leq C n^{δ - 1/6}$ ) for every $δ > 0$ . <a href="#NPV2011-thm1-powerlaw">[NPV2011-thm1-powerlaw]</a>

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, $exp (- c lo g_{*} n)$ , attributed to Peres–Solomyak. <a href="#NPV2011-exp-logstar">[NPV2011-exp-logstar]</a>

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). <a href="#NPV2011-remark-n-1-4">[NPV2011-remark-n-1-4]</a>

Wikipedia page on Favard length

References

<a id="BV2010"></a>[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
<a id="BV2010-thm1-lb-logn-over-n"></a>[BV2010-thm1-lb-logn-over-n] loc: arXiv v1 PDF p.3, Theorem 1 (display (1.2)) quote: “There exists $c > 0$ such that $Fav (K_{n}) \geq c \frac{l o g n}{n}$ for all $n \in N$ .”

<a id="NPV2011"></a>[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
<a id="NPV2011-besicovitch-open-decay"></a>[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.”
<a id="NPV2011-exp-logstar"></a>[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 $exp (- c lo g_{*} n)$ , due to Peres and Solomyak.”
<a id="NPV2011-def-fav"></a>[NPV2011-def-fav] loc: arXiv v1 PDF p.2, equation (1.1) quote: “ $Fav (E) = \frac{1}{π} \int_{0}^{π} ∣ Proj R_{θ} E ∣ d θ$ .”
<a id="NPV2011-buffon-interpretation"></a>[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 $E$ ”
<a id="NPV2011-Kn-4n-squares"></a>[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 $4^{- n}$ .”
<a id="NPV2011-thm1-powerlaw"></a>[NPV2011-thm1-powerlaw] loc: arXiv v1 PDF p.3, Theorem 1 quote: “For every $δ > 0$ , there exists $C > 0$ such that $Fav (K_{n}) \leq C n^{δ - 1/6}$ for all $n \in N$ .”
<a id="NPV2011-remark-n-1-4"></a>[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.”

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.

Current state

The problem