Kronecker Sequences with Many Distances

The three gap theorem states that for any alpha is an element of R and N is an element of N, the number of different gaps between consecutive n alpha( mod 1) for n is an element of{1, horizontal ellipsis ,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 over bar pd so that g over bar p1=3. Haynes and Marklof (2021) showed that each example with arbitrary alpha and N gives a generic lower bound, and that g over bar 22=5 and g over bar 2d <=sigma d+1 where sigma d is the kissing number. They gave an example showing g over bar 23 >= 7. Our examples that show g over bar 23 >= 9 and also g over bar 24 >= 11, g over bar 25 >= 13 and g over bar 26 >= 14. Haynes and Ramirez (2021) showed that g over bar infinity d <= 2d+1 and that this is sharp for d <= 3. We provide a numerical example to show g over bar infinity 4 >= 15, and a proof that g over bar infinity d >= 2d-1+1 in general. Results for p=infinity and sigma d imply that g over bar pd depends on p for d >= 11 and we conjecture this for d >= 4. For d <= 3 we expect that g over bar 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 over bar p2 >= 5 and for d = 3 we provide numerical examples suggesting that g over bar p3 >= 9.