Alan Haynes - “Gaps problems in Diophantine approximation and extremal problems about lattices“

The talk “Gaps problems in Diophantine approximation and extremal problems about lattices“ by Alan Haynes on the Moscow Conference on Combinatorics and Applications at MIPT. Annotation: The three distance theorem states that, if x is any real number and N is any positive integer, the points x, 2x, … , Nx modulo 1 partition the unit interval into component intervals having at most 3 distinct lengths. We present a generalization of this problem for rotations on higher dimensional tori, and we explain how to reformulate it as a problem about lattices in Euclidean space. For the two-dimensional torus, we are able to prove a five distance theorem, which is best possible. In higher dimensions we do not know the best possible bounds. Determining them can be viewed as an extremal problem in discrete geometry. The best currently known bounds are given by the kissing number, but these are likely far from the truth. The full schedule of the conference -