A Five Distance Theorem for Kronecker Sequences

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, ..., N alpha mod 1. In this paper, we consider a natural generalization of the three-distance theorem to the higher-dimensional Kronecker sequence (alpha) over right arrow2 (alpha) over right arrow, ..., N (alpha) over right arrow modulo an integer lattice. We prove that in 2D, there are at most five values that can arise as a distance between nearest neighbors, for all choices of (alpha) over right arrow and N. Furthermore, for almost every (alpha) over right arrow, 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 3D, 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 generalization of the gap length in 1D. 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 (1) unbounded for almost all (alpha) over right arrow and (2) bounded for (alpha) over right arrow that satisfy certain Diophantine conditions.

Automated summary

This paper extends the three-distance theorem and provides a generalization for higher-dimensional Kronecker sequences. It proves that in 2D, there can be at most five possible distances between nearest neighbors, and conjectures similar upper bounds for higher dimensions. Additionally, it studies the number of possible distances from a point to its nearest neighbor in a restricted cone of directions and provides results under different conditions.