Dynamical Systems Seminar

The celebrated Three-Gap Theorem states that, if one places first N elements of the Kronecker sequence {nx}, n=1,…, N, on a unit circle, then distances between consecutive points take no more than three distinct values. I will talk about the higher-dimensional version of this theorem. Recently, Haynes and Marklof solved the problem in two dimensions by showing that the number of gaps in a two-dimensional Kronecker sequence is no greater than 5. I will show how this problem is connected to a general sphere packing problem and explain several new bounds on the number of gaps in all dimensions confirming, in particular, a weak version of the conjecture of Haynes and Marklof in three dimensions.