Nine-distance theorem and growth of best-approximation denominators

2026/08/02 by Nikita Shulga
Mathematics · #math.NT #math.CO #msc:11J13 #msc:11K31 #msc:52C17

paper · pdf

15 pages

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first N orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we prove that on a flat d-dimensional torus the number of such distances is at most 2^d+1. The main tool is a new growth theorem for the denominators q₁<q₂<⋯ of best simultaneous approximations in a d-dimensional inner-product space, which is of independent interest. We prove that, whenever q_n+2^d is defined, either q_n+2^d≥2q_n+1, or the indices 1,…,2^d can be partitioned into disjoint pairs \j,k\, j<k, such that q_n+k=q_n+q_n+j. In particular, q_n+2^d≥ min\2q_n+1,q_n+q_n+2^d-1\≥ q_n+q_n+1.

Citations

• A five distance theorem for Kronecker sequences
• The three gap theorem and the space of lattices
• Simultaneous two-dimensional best Diophantine approximations in the Euclidean norm
• The Three Gap Theorem and Riemannian Geometry
• Sum-free sets in abelian groups