| In recent years, progress has been made on Buffon's needle problem, in which one considers a subset of the plane and asks how likely "Buffon's needle" - a long, straight needle with independent, uniform distributions on its position and orientation - is to intersect said set. The case in which the set is a small neighborhood of a one-dimensional unrectifiable Cantor-like set has been considered in recent years, and progress has been made, motivated in part by connections to analytic capacity [25].;Call the set E, the radius of the neighborhood epsilon, and the neighborhood Eepsilon. Then in some special cases [5][13][18], it has been confirmed that Buffon's needle intersects Eepsilon with probability at most C|log epsilon| -p, for p > 0 small enough, C > 0 large enough. In the special case of the so-called "four corner" Cantor set and Sierpinski's gasket, the lower bound Cloglog3 log3 is known [3], replacing the previously-known lower bound Clog3 which is good for more general one-dimensional self-similar sets.;In addition, the stronger lower bounds are still good if one "bends the needle" into the shape of a long circular arc, or "Buffon's noodle." The radius one uses can be as small as |log epsilon|epsilon0, for any epsilon 0 > 0, with the constant C depending on epsilon 0 [6]. It is unknown whether this condition or anything like it is necessary.;Work continues on generalizing the upper bound results. |
