Multi-dimensional Kronecker Sequences with a Small Number of Gap Lengths

TL;DR

This paper explores multi-dimensional Kronecker sequences, showing that in dimensions 2 and 3, it is possible to construct sequences with a surprisingly small number of distinct nearest neighbor distances, contrasting with generic cases.

Contribution

The paper constructs specific multi-dimensional Kronecker sequences in dimensions 2 and 3 that have a minimal number of gap lengths, using simple continued fraction arguments.

Findings

Constructed sequences with few gap lengths in dimensions 2 and 3.

Contrasted generic maximal gap lengths with special low-gap sequences.

Used continued fraction theory for the construction and analysis.

Abstract

Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum metric. It was proved that a generic multi-dimensional Kronecker attains the maximal possible number of different gap lengths for every sub-exponential subsequence. We mirror this result in dimension $d \in \left\{ 2, 3 \right\}$ by constructing Kronecker sequences which have a surprisingly low number of different nearest neighbor distances for infinitely $N \in \mathbb{N}$. Our proof relies on simple arguments from the theory of continued fractions.