Estimates from below of the Buffon noodle probability for undercooked noodles

TL;DR

This paper investigates whether bending needles into certain shapes, called noodles, affects the probability estimates of intersecting a fractal set, and proves that specific classes of curved noodles still maintain lower probability bounds.

Contribution

It introduces the concept of undercooked noodles, extending Favard length estimates to curved shapes like circular arcs, and establishes conditions under which these noodles remain undercooked.

Findings

Circular arcs with radius at least 4^{n/5} are undercooked noodles.

Lower bounds for noodle intersection probabilities are preserved for certain curved shapes.

The study generalizes Favard length estimates to more complex geometric objects.

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.