Kronecker Sequences with Many Distances

The three gap theorem states that for any α∈R and N∈N, the number of different gaps between consecutive nα (mod 1) for n∈1, …, N is at most 3. Biringer and Schmidt (2008) instead considered the distance from each point to its nearest neighbor, generalizing to higher dimensions. We denote the maximum number of distances in Td using the p-norm by g¯pd so that g¯p1=3. Haynes and Marklof (2021) showed that each example with arbitrary α and N gives a generic lower bound, and that g¯22=5 and g¯2d≤σd+1 where σd is the kissing number. They gave an example showing g¯23≥7. Our examples that show g¯23≥9 and also g¯24≥11, g¯25≥13 and g¯26≥14. Haynes and Ramirez (2021) showed that g¯∞d≤2d+1 and that this is sharp for d≤3. We provide a numerical example to show g¯∞4≥15, and a proof that g¯∞d≥2d−1+1 in general. Results for p=∞ and σd imply that g¯pd depends on p for d≥11 and we conjecture this for d≥4. For d≤3 we expect that g¯pd=3, 5, 9 for d=1, 2, 3 respectively, independent of p. For d = 1 this is trivial, for d = 2 we show that g¯p2≥5 and for d = 3 we provide numerical examples suggesting that g¯p3≥9.