Submit progress

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

The statement

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

\mathrm{Fav}(E)\ :=\ \frac{1}{\pi}\int_0^\pi \lvert \mathrm{Proj}\,R_\theta E\rvert\, d\theta,

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

K_n\ :=\ C_n \times C_n.

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

C_{60}\ :=\ \alpha_{\mathrm{Fav}}\ :=\ \sup\Bigl\{\alpha\ge 0:\ \exists C>0\ \text{such that}\ \mathrm{Fav}(K_n)\le C\,n^{-\alpha}\ \text{for all}\ n\in\mathbb{N}\Bigr\}.

The best established range currently is

\frac{1}{6}\ \le\ \alpha_{\mathrm{Fav}}\ \le\ 1.

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

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.