Is there a quick algorithm to determine whether a finite set A⊂Z tiles the integers by translations? Can we estimate the minimal period of such a tiling? What is the relation between tiling and harmonic-analytic properties of a set? In a different direction: let E be a purely unrectifiable self-similar planar set of Hausdorff dimension 1. What can we say about the asymptotic behaviour of the length of an average (with respect to angle) linear projection of its finite iterates En as n→∞?

These seemingly unrelated questions all lead to cyclotomic divisibility. The s-th cyclotomic polynomial Φs is the minimal polynomial of the s-th primitive roots of unity. A key issue in the problems named above is understanding under what circumstances a polynomial may be divisible by many such cyclotomic polynomials simultaneously. In this talk I will discuss the basic concepts involved, open questions, and some of the progress made in the last few years. Based in part on my joint work with Benjamin Bruce, Itay Londner, and Caleb Marshall.