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(mod1) for n ∈ {1, ..., N} is at most 3. Biringer and Schmidt (2008) instead considered the distance from each point to its nearest neighbour, generalising to higher dimensions. We denote the maximum number of distances in T d using the p-norm by ¯g d p so that ¯g 1 p = 3. Haynes and Marklof (2021) showed that each example with arbitrary α and N gives a generic lower bound, and that ¯g 2 2 = 5 and ¯g d 2 ≤ σd + 1 where σd is the kissing number. They gave an example showing ¯g 3 2 ≥ 7. Our examples that show ¯g 3 2 ≥ 9 and also g¯ 4 2 ≥ 11, ¯g 5 2 ≥ 13 and ¯g 6 2 ≥ 14. Haynes and Ramirez (2021) showed that ¯g d∞ ≤ 2 d + 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∞ ≥ 2 d−1 + 1 in general. Results for p = ∞ and σd imply that ¯g d p depends on p for d ≥ 11 and we conjecture this for d ≥ 4. For d ≤ 3 we expect that ¯g d p = {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 2 p ≥ 5 and for d = 3 we provide numerical examples suggesting that ¯g 3 p ≥ 9.