Sharp decay rate of the Favard length of the four-corner Cantor set

The Favard length of a planar set is the average length of its orthogonal projections and equals, up to normalization, the probability that Buffon's needle dropped at random hits the set. For the n-th generation $K_n$ of the four-corner (1/4) Cantor set, Besicovitch's projection theorem forces the Favard length to tend to zero, but its sharp decay rate is unknown. The problem, posed in Bishop and Peres' chapter on Besicovitch–Kakeya sets, is to determine this rate. The best known bounds are the lower bound $\operatorname{Fav}(K_n)\ge c\,\log n/n$ (Bateman–Volberg, 2010) and the upper bound $\operatorname{Fav}(K_n)\le C_\varepsilon n^{-1/5+\varepsilon}$ for every $\varepsilon>0$ (Marshall, 2025, improving Nazarov–Peres–Volberg's $n^{-\tau}$, $\tau<1/6$), leaving a large gap; the conjectured truth, by analogy with random four-corner sets whose Favard length decays exactly like $1/n$, is $n^{-1+o(1)}$.

Background

Let $C_0=[0,1]$, and let $C_{n+1}$ be obtained from $C_n$ by keeping, inside each component interval, only the two subintervals of a quarter of the length situated at the ends (removing the middle half). The limit $C=\bigcap_n C_n$ is the middle-half or 1/4 Cantor set, a set of Hausdorff dimension $1/2$; $C_n$ is a union of $2^n$ intervals of length $4^{-n}$. The four-corner Cantor set is the planar product $K=C\times C$; its n-th generation $K_n=C_n\times C_n$ is a union of $4^n$ closed squares of side $4^{-n}$. The limit set $K$ has Hausdorff dimension $1$ and is purely unrectifiable, meaning it intersects every curve of finite length in length zero. It is also called the Garnett set, after John Garnett used it in complex analysis as an example of a set of positive length and zero analytic capacity, analytic capacity being the measure of removability for bounded analytic functions that appears in the Painlevé problem.

For a compact set $E\subset\mathbb{R}^2$, the Favard length is the angular average of the lengths of its orthogonal projections, $\operatorname{Fav}(E)=\frac{1}{\pi}\int_0^{\pi}\bigl|\operatorname{proj}_{\theta}(E)\bigr|\,d\theta,$ where $|\cdot|$ denotes one-dimensional Lebesgue measure and $\operatorname{proj}_{\theta}$ is the orthogonal projection onto a line making angle $\theta$ with the horizontal axis. Up to normalization, $\operatorname{Fav}(E)$ is the probability that Buffon's needle — a long line segment dropped at a uniformly random position and angle — meets $E$; for sets built from squares of side $4^{-n}$ it is also comparable to the probability that a random line meets the set. By the Besicovitch projection theorem, a purely unrectifiable set of finite length projects to measure zero in almost every direction, so $\operatorname{Fav}(K_n)\to 0$ as $n\to\infty$. The question, posed in the Besicovitch–Kakeya chapter of Bishop and Peres' book Fractals in Probability and Analysis, is how fast.

The difficulty is that the pointwise decay of $|\operatorname{proj}_{\theta}(K_n)|$ in $n$ is extremely non-uniform in the direction $\theta$: in the coordinate directions the projection length is $2^{-n}$, there is a dense set of directions with exponential-in-$n$ decay, and there is a dense set of measure zero in which the projections of the limit set have positive length, phenomena governed by the tiling arithmetic of the digit set (see Lagarias and Wang, and the survey of Łaba). Any bound on the average must therefore control the measure of exceptional direction sets at every scale simultaneously.

The known bounds are as follows. Mattila proved a general lower bound $\operatorname{Fav}(E_n)\ge C/n$ for self-similar dimension-one Cantor iterations, which Bateman and Volberg improved for the four-corner set to $\operatorname{Fav}(K_n)\ge c\,\log n/n$. On the upper-bound side, Peres and Solomyak first obtained a general quantitative bound of the form $\exp(-c\log_* n)$ (where $\log_* n$ is the iterated logarithm); Nazarov, Peres, and Volberg then proved the first power-law bound $\operatorname{Fav}(K_n)\le C_\tau n^{-\tau}$ for every $\tau<1/6$ using Fourier-analytic methods, initiating the modern line of work; Bond, Łaba, and Volberg and subsequently Łaba and Marshall developed number-theoretic techniques based on vanishing sums of roots of unity — a vanishing sum of roots of unity is a sum of complex numbers of finite multiplicative order that equals zero, and such sums constrain how the projected copies of the digit set can overlap — for the wider class of rational product Cantor sets, that is, products of one-dimensional Cantor sets generated by integer digit sets in a fixed integer base; Marshall most recently improved the four-corner upper bound to $\operatorname{Fav}(K_n)\le C_\varepsilon n^{-1/5+\varepsilon}$ for every $\varepsilon>0$. For contrast, Peres and Solomyak proved that for random four-corner sets the expected Favard length decays exactly like $C/n$, and this sharp random theory has recently been extended to broad classes of planar random fractals by Chang, Shmerkin, and Suomala. Łaba's survey records the expectation that the optimal deterministic upper bound should be $C_\varepsilon n^{-1+\varepsilon}$ for every $\varepsilon>0$, with the $n^{\varepsilon}$ slack unavoidable in view of the $\log n/n$ lower bound.

Problem Statement

Let $C_n$ denote the n-th generation of the middle-half (1/4) Cantor set in $[0,1]$, obtained from $[0,1]$ by iterating the removal of the open middle half of each component interval, so that $C_n$ is a union of $2^n$ closed intervals of length $4^{-n}$, and let $K_n=C_n\times C_n$ be the n-th generation of the four-corner Cantor set, a union of $4^n$ squares of side $4^{-n}$. Let $\operatorname{Fav}(K_n)=\frac{1}{\pi}\int_0^{\pi}\bigl|\operatorname{proj}_{\theta}(K_n)\bigr|\,d\theta$ be its Favard length, the average length of its orthogonal projections, equivalently (up to normalization) the probability that Buffon's needle dropped at random meets $K_n$. Determine the sharp asymptotic decay rate of $\operatorname{Fav}(K_n)$ as $n\to\infty$, i.e. an explicit rate $f$ with $c\,f(n)\le\operatorname{Fav}(K_n)\le C\,f(n)$ for large $n$, including any logarithmic corrections. In particular, close the gap between the best known lower bound $\operatorname{Fav}(K_n)\ge c\,\log n/n$ and the best known upper bound $\operatorname{Fav}(K_n)\le C_\varepsilon n^{-1/5+\varepsilon}$ for every $\varepsilon>0$, and decide whether the conjectured behavior $\operatorname{Fav}(K_n)=n^{-1+o(1)}$ holds.

References

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