Upper and lower bounds on the rate of decay of the Favard curve length for the four-corner Cantor set

Abstract

The Favard length of a subset of the plane is defined as the average of its orthogonal projections. This quantity is related to the probabilistic Buffon needle problem; that is, the Favard length of a set is proportional to the probability that a needle or a line that is dropped at random onto the set will intersect the set. If instead of dropping lines onto a set, we drop fixed curves, then the associated Buffon curve probability is proportional to the so-called Favard curve length. As we show in our companion paper, a Besicovitch generalized projection theorem still holds in the setting where lines are replaced by curves. Consequently, the Favard curve length of any purely unrectifiable set is zero. Since the four-corner Cantor set is a compact, purely unrectifiable $1$-set with bounded, non-zero Hausdorff measure, then its Favard curve length equals zero. In this article, we estimate upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set. Our techniques build on the ideas that have been previously used for the classical Favard length.