Kronecker Sequences with Many Distances

Abstract

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 T d using the p-norm by g ¯ p d so that g ¯ p 1 = 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 ¯ 2 d ≤ σ d + 1 where σd is the kissing number. They gave an example showing g ¯ 2 3 ≥ 7 . Our examples that show g ¯ 2 3 ≥ 9 and also g ¯ 2 4 ≥ 11 , g ¯ 2 5 ≥ 13 and g ¯ 2 6 ≥ 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 ¯ p d depends on p for d ≥ 11 and we conjecture this for d ≥ 4 . For d ≤ 3 we expect that g ¯ p d = { 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 ¯ p 2 ≥ 5 and for d = 3 we provide numerical examples suggesting that g ¯ p 3 ≥ 9 .