A five distance theorem for Kronecker sequences

TL;DR

This paper generalizes the three distance theorem to higher dimensions, establishing upper bounds on the number of possible distances between points in Kronecker sequences modulo lattices, with results varying by dimension and Diophantine properties.

Contribution

It proves a five-distance upper bound in two dimensions for all sequences and almost surely attains this bound, extending the three distance theorem to higher dimensions with explicit bounds.

Findings

In 2D, at most 5 distances occur for all sequences.

Almost every sequence exhibits 5 distances infinitely often in 2D.

Higher dimensions have explicit but less precise bounds, e.g., 13 in 3D.

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.