2008/11/09 by Matthew Bond, Alexander Volberg, Bond, Matthew +1
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Computability, Logic, AI Algorithms #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.0811.1302
Let \Cantn be the n-th generation in the construction of the middle-half Cantor set. The Cartesian square \Kn of \Cantn consists of 4n squares of side-length 4-n. The chance that a long needle thrown at random in the unit square will meet \Kn is essentially the average length of the projections of \Kn, also known as the Favard length of \Kn. A result due to Bateman and Volberg \citeBV shows that a lower estimate for this Favard length is c (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 (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 rn. We will show that if rn ≥ 4(n)/(5), then the circular arcs are undercooked noodles.