A five distance theorem for Kronecker sequences

Abstract

The three distance theorem (also known as the three gap theorem or Steinhaus problem) states that, for any given real number $\alpha$ and integer $N$, there are at most three values for the distances between consecutive elements of the Kronecker sequence $\alpha, 2\alpha,\ldots, N\alpha$ mod 1. In this paper we consider a natural generalisation of the three distance theorem to the higher dimensional Kronecker sequence $\vec\alpha, 2\vec\alpha,\ldots, N\vec\alpha$ modulo an integer lattice. We prove that in two dimensions there are at most five values that can arise as a distance between nearest neighbors, for all choices of $\vec\alpha$ and $N$. Furthermore, for almost every $\vec\alpha$, five distinct distances indeed appear for infinitely many $N$ and hence five is the best possible general upper bound. In higher dimensions we have similar explicit, but less precise, upper bounds. For instance in three dimensions our bound is 13, though we conjecture the truth to be 9. We furthermore study the number of possible distances from a point to its nearest neighbor in a restricted cone of directions. This may be viewed as a generalisation of the gap length in one dimension. For large cone angles we use geometric arguments to produce explicit bounds directly analogous to the three distance theorem. For small cone angles we use ergodic theory of homogeneous flows in the space of unimodular lattices to show that the number of distinct lengths is (a) unbounded for almost all $\vec\alpha$ and (b) bounded for $\vec\alpha$ that satisfy certain Diophantine conditions.