Estimates from below of the Buffon noodle probability for undercooked noodles

Abstract

Let $\Cant_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $\K_n$ of $\Cant_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $\K_n$ is essentially the average length of the projections of $\K_n$, also known as the Favard length of $\K_n$. A result due to Bateman and Volberg \cite{BV} shows that a lower estimate for this Favard length is $c \frac{\log n}{n}$. We may bend the needle at each stage, giving us what we will call a noodle, and ask whether the uniform lower estimate $c \frac{\log n}{n}$ still holds for these so-called Buffon noodle probabilities. If so, we call the sequence of noodles undercooked. We will define a few classes of noodles and prove that they are undercooked. In particular, we are interested in the case when the noodles are circular arcs of radius $r_n$. We will show that if $r_n \geq 4^{\frac{n}{5}}$, then the circular arcs are undercooked noodles.