For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this thesis.}\end{figure}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.Call the set E, the radius of the neighborhood ε, and the neighborhood Eε. Then in some special cases, it has been confirmed that Buffon's needle intersects Eε with probability at most C|logε|-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 C*log|logε|/|logε| is known, replacing the previously-known lower bound C/|logε| 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 |logε|ε0$, for any ε0, with the constant C depending on ε0. It is unknown whether this condition or anything like it is necessary.Work continues on generalizing the upper bound results.
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