NTAG Seminar: Kronecker sequences with many distances

The 3 gap theorem states that for any real alpha and natural N, the sequence n alpha (mod 1) for 1<=n<=N has at most three different sized gaps. Biringer and Schmidt (2008) generalise this to higher dimensions by replacing gaps by the distance from each point to its nearest neighbour. Haynes and Marklof (2021) give a 5 distance theorem in dimension d=2 using Euclidean distance (p=2). Haynes and Ramirez (2021) consider the maximum distance (p=infinity), and find 5 distances for d=2 and 9 distances for d=3. In both cases there is an exponential upper bound in higher dimensions. I will describe recent work giving lower bounds for d<=6 for p=2, all d for p=infinity and d=2 for all p, which show that the maximum number of distances varies with p for d>=11.