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The Favard length of a subset of the plane is defined as the average of itsorthogonal projections. This quantity is related to the probabilistic Buffonneedle problem; that is, the Favard length of a set is proportional to theprobability that a needle or a line that is dropped at random onto the set willintersect the set. If instead of dropping lines onto a set, we drop fixedcurves, then the associated Buffon curve probability is proportional to theso-called Favard curve length. As we show in our companion paper, a Besicovitchgeneralized projection theorem still holds in the setting where lines arereplaced by curves. Consequently, the Favard curve length of any purelyunrectifiable set is zero. Since the four-corner Cantor set is a compact,purely unrectifiable $1$-set with bounded, non-zero Hausdorff measure, then itsFavard curve length equals zero. In this article, we estimate upper and lowerbounds for the rate of decay of the Favard curve length of the four-cornerCantor set. Our techniques build on the ideas that have been previously usedfor the classical Favard length.

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