Parasitic lampreys and other problems

Myself, Izabella Laba, and Alexander Volberg have kicked off the semester with a new paper, Buffon’s needle estimates for rational product sets. As we try to generalize old results without simultaneously weakening them, we run into a new obstacle – the parasitic lamprey.

Not that lamprey.

What we mean is the Linear Multi-Polygon Relation (LMPRe). That is, a vanishing sum of roots of unity. A very old subject that is not entirely understood, one could call it a “living fossil” of sorts.

In particular… Well, you’ll have to read the paper, but in any case, they are legitimate obstacles to Buffon’s needle problem. But as we look for ways to “sidestep parasitic lampreys”, we lose more and more freedom to ignore terms in our equations, as the parasitic lampreys “suck away” large chunks of our “good sets”. Eventually, the existing method will either break down, or it will succeed because of our new insights into cyclotomic divisors of {0,1} polynomials. I’m hoping for the latter, naturally. But we will see.

The debate over the appropriateness of the terminology is underway here.