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Nine-distance theorem and growth of best-approximation denominators

This paper proves the Haynes-Marklof conjecture by establishing a nine-distance theorem for Kronecker sequences on flat three-tori and a general $2^d+1$ bound for $d$ -dimensional tori, achieved through a novel growth theorem for denominators of best simultaneous approximations.

Original authors: Nikita Shulga

Published 2026-08-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nikita Shulga

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic Dance of Points and the Mystery of Gaps

Imagine you are standing in a giant, perfectly smooth room that wraps around itself like a video game world. If you walk in a straight line, you eventually return to where you started, but slightly shifted. Now, imagine dropping a pebble at regular intervals as you walk. In a flat, one-dimensional hallway, these pebbles would create gaps of different sizes between them. A famous old puzzle, known as the Three-Distance Theorem, revealed a surprising secret: no matter how many pebbles you drop, there can only ever be three different sizes of gaps between them. It's as if the universe has a strict rulebook for how these points can arrange themselves.

But what happens when we leave the hallway and enter a multi-dimensional room? What if we are walking on a 3D torus, or even a 100-dimensional space? This is the realm of "Kronecker sequences," a branch of mathematics that studies how points distribute themselves on shapes that loop back on themselves. Mathematicians have long wondered: as we add more dimensions, does the number of possible gap sizes explode? Or is there still a hidden limit, a cosmic cap on the chaos? Understanding this isn't just about abstract shapes; it helps us understand the fundamental geometry of space, how numbers interact, and how order emerges from randomness in high-dimensional systems.

The Nine-Distance Discovery

In this paper, mathematician Nikita Shulga tackles the question of how many different gap sizes can exist when points are scattered on a flat, multi-dimensional torus. The paper proves a "Nine-Distance Theorem" for three-dimensional spaces and establishes a general rule for any number of dimensions.

The Main Finding
Shulga proves that if you scatter $N$ points on a flat, three-dimensional torus (a 3D space that loops back on itself), the number of distinct distances between nearest neighbors is never more than 9. This confirms a conjecture made by Haynes and Marklof. Furthermore, the paper establishes a universal rule for any dimension $d$ : the number of distinct gap sizes is at most $2d + 1$ . So, in a 1D hallway, the limit is 3; in a 2D plane, it's 5; in our 3D world, it's 9; and in a 4D space, it would be 17.

The "Best Approximation" Secret Weapon
To crack this code, Shulga didn't just look at the gaps directly. Instead, he looked at the "denominators of best simultaneous approximations." In simpler terms, imagine trying to find the best whole-number steps to approximate a specific, irrational walking rhythm. Some steps get you very close to a perfect landing spot, while others miss by a lot. These "best" steps have specific sizes (denominators). Shulga discovered a new growth rule for these denominators: they must grow fast enough, or they must follow a very specific additive pattern. He proved that for every $2d$ steps in this sequence, the next denominator is either at least double the previous one, or it can be built by adding two earlier denominators together. This "growth theorem" acts like a speed limit for how slowly these numbers can grow, which in turn limits how many different gap sizes can appear.

What is Ruled Out?
The paper explicitly rules out the idea that the number of gap sizes could be higher than $2d + 1$ . Before this work, mathematicians had a looser bound for 3D spaces, suggesting the limit might be 13. Shulga's proof tightens this to 9, showing that the "13" scenario is impossible. The paper also clarifies that while some previous results relied on specific norms (like the "supremum norm" used in computer science), this new bound holds true for any standard geometric distance (inner-product norm), making the result much more robust.

How Sure Are We?
This is a proven mathematical theorem, not a simulation or a guess. The author provides a rigorous, step-by-step proof using logic and geometry. The paper also confirms that the number 9 is "optimal" or "sharp" for three dimensions. This means there is a specific, real example (found by Dettmann) where exactly 9 different distances actually occur. You cannot lower the limit to 8, because 9 is achievable. Similarly, the paper shows that the bounds for 1D (3 distances) and 2D (5 distances) are also the best possible, matching known examples.

The Takeaway
The paper concludes that the universe of these point patterns is far more orderly than it appears. No matter how complex the dimension, the chaos of gaps is tamed by a simple formula: $2d + 1$ . For our 3D world, the magic number is 9. This result unifies previous findings and provides a powerful new tool—a growth theorem for denominators—that mathematicians can use to solve other problems in number theory and geometry. It turns out that even in the highest dimensions, there is a strict, predictable rhythm to how points arrange themselves.

Technical Summary: Nine-Distance Theorem and Growth of Best-Approximation Denominators

Problem Statement
The paper addresses the higher-dimensional analogue of the classical three-distance theorem (Steinhaus conjecture). In one dimension, the theorem states that the gaps between the first $N$ points of a Kronecker sequence $\{n\alpha \pmod 1\}_{n=1}^N$ on the circle take at most three distinct values. The author investigates the number of distinct nearest-neighbour distances, denoted $g_N(\alpha, L)$ , generated by the first $N$ orbit points of a Kronecker sequence on a flat $d$ -dimensional torus $V/L$ .

Prior to this work, Haynes and Marklof established an upper bound of 5 for the planar case ( $d=2$ ) and proved it is sharp. For dimension $d=3$ , they provided a bound of 13 and conjectured that the optimal universal bound is 9. Dettmann had previously demonstrated an example on the cubic three-torus where 9 distinct distances occur, suggesting 9 is the sharp bound for $d=3$ . The central problem is to determine the optimal dimension-uniform upper bound for $g_N(\alpha, L)$ and to settle the conjecture for $d=3$ .

Methodology
The primary tool developed is a new growth theorem for the denominators of best simultaneous approximations in a $d$ -dimensional inner-product space. Let $q_1 < q_2 < \dots$ be the sequence of integers where the distance $\rho(q) = \text{dist}(q\alpha, L)$ attains a new strict minimum.

The author proves a structural theorem (Theorem 1.4) regarding the growth of these denominators. Specifically, for $M=2d$ , whenever $q_{n+M}$ is defined, one of two conditions must hold:

Rapid Growth: $q_{n+2d} \ge 2q_{n+1}$ .
Additive Structure: The set of indices $\{1, \dots, 2d\}$ can be partitioned into disjoint pairs $\{j, k\}$ with $j < k$ such that $q_{n+k} = q_n + q_{n+j}$ .

This result is derived using a combination of:

Sum-free set theory: Applying a classical bound on sum-free subsets in $\mathbb{F}_2^{d+1}$ (Lemma 3.1) to the parity classes of the coordinate vectors of the best approximations.
Geometric identities: Utilizing a generalized parallelogram law (Lemma 3.2) to derive contradictions if the additive structure fails, thereby forcing the rapid growth condition.

The paper also revisits the relationship between nearest-neighbour distances and best-approximation denominators, following the framework of Chevallier and Haynes-Marklof. It establishes that the number of distinct distances is bounded by the number of best-approximation denominators falling within a specific interval relative to $N$ .

Key Results

The Nine-Distance Theorem (Theorem 1.2): For any three-dimensional real inner-product space, full-rank lattice, and rotation vector, the number of distinct nearest-neighbour distances among the first $N$ orbit points is at most 9. This confirms the conjecture of Haynes and Marklof.
Dimension-Uniform Bound (Theorem 1.1): For any dimension $d \ge 1$ , the number of distinct distances satisfies $g_N(\alpha, L) \le 2d + 1$ .
Sharpness: The bound is shown to be sharp for dimensions $d=1, 2,$ and $3$. The $d=3$ case is sharp due to Dettmann's example yielding exactly 9 distances.
Growth Rate Corollary (Corollary 3.3): The author derives a lower bound for the exponential growth rate of the best-approximation denominators. For $d \ge 2$ , $\liminf_{n \to \infty} q_n^{1/n} \ge 2^{1/(2d-1)}$ . This improves upon previous lower bounds for $d=2$ (improving Ermakov's bound from $\approx 1.228$ to $\approx 1.260$ ).

Comparison with Existing Bounds
The paper compares its new bound ( $2d+1$ ) against the bound derived from the kissing number (or contact number) $\sigma_d$ , which yields $g_N \le \sigma_d + 1$ .

For $d=3$ , the kissing number bound is 13, while the new bound is 9.
For $d=8$ , the new bound is 17, whereas the kissing number bound (using the strict inequality definition $\sigma^>_d$ ) is 240.
The author notes that while the kissing number bound is historically significant and implicit in earlier work combining Chevallier's and Romanov's results, the new $2d+1$ bound is superior for small dimensions and provides a dimension-uniform linear bound, whereas the kissing number bound grows exponentially.

Significance
The paper claims to settle the specific conjecture regarding the nine-distance theorem in dimension three and provides a general, dimension-uniform upper bound for the number of distinct nearest-neighbour distances in flat tori. The significance lies in the introduction of a new growth theorem for best-approximation denominators that applies to every inner-product norm (not just the supremum norm) and includes a precise description of the exceptional case where rapid growth does not occur. This structural insight allows for a tighter control over the distribution of orbit points than previously available via geometric lattice arguments alone. The results are presented as definitive for the stated problem, with the bounds proven to be optimal in the first three dimensions.

The Cosmic Dance of Points and the Mystery of Gaps

The Nine-Distance Discovery

Technical Summary: Nine-Distance Theorem and Growth of Best-Approximation Denominators

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