The celebrated Three-Gap Theorem states that, if one places first N
elements of the Kronecker sequence {nx}, n=1,…, N, on a unit circle,
then distances between consecutive points take no more than three
distinct values. I will talk about the higher-dimensional version of
this theorem. Recently, Haynes and Marklof solved the problem in two
dimensions by showing that the number of gaps in a two-dimensional
Kronecker sequence is no greater than 5. I will show how this problem
is connected to a general sphere packing problem and explain several
new bounds on the number of gaps in all dimensions confirming, in
particular, a weak version of the conjecture of Haynes and Marklof in
three dimensions.
|