arXiv:2608.01443v1

Nine-distance theorem and growth of best-approximation denominators

Nikita Shulga

math.NTmath.CO11J1311K3152C17

Paper metadata

Original submission: August 2, 2026
Claimed MSC: 11J13, 11K31, 52C17
arXiv categories: math.NT, math.CO
Paper license: CC BY 4.0 ↗
Comments: 15 pages

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 $q_1<q_2<\cdots$ 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 $q_{n+2^d}\ge2q_{n+1}$ , or the indices $1,\ldots,2^d$ can be partitioned into disjoint pairs $\{j,k\}$ , $j<k$ , such that $q_{n+k}=q_n+q_{n+j}$ . In particular, $q_{n+2^d}\ge \min\{2q_{n+1},q_n+q_{n+2^{d-1}}\}\ge q_n+q_{n+1}.$

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The denominator-growth dichotomy, its growth consequence, the $2^d+1$ distance bound, the nine-distance theorem, and the sharpness claims checked are correct.

Theorem 1.4Correct

Growth dichotomy for best-approximation denominators

Pages 3–9 · Theorem 1.4 · arXiv:2608.01443v1

With $M=2^d$ , the parity classes of the $M$ later best approximants form a maximum-size sum-free subset of $\mathbb F_2^{d+1}$ in the exceptional regime. Its equality structure supplies a fixed-point-free pairing. The parallelogram-law argument then forces $q_{n+k}=q_n+q_{n+j}$ for each pair, proving both alternatives and the consequent recurrence.

Theorems 1.1 and 1.2Correct

The $2^d+1$ bound and the nine-distance theorem

Pages 2 and 11–13 · Theorems 1.1–1.2 and their proof · arXiv:2608.01443v1

Proposition 5.1 identifies the distinct nearest-neighbour distances with the record minima whose denominators lie in $(\lfloor N/2\rfloor,N-1]$ , plus one initial value. Theorem 1.4 prevents more than $2^d$ record denominators from lying in that interval, giving $g_N(\boldsymbol\alpha,\mathcal L)\leq2^d+1$ and hence $g_N\leq9$ when $d=3$ . The finite-order case is covered separately and correctly.

Corollary 5.2Correct

Sharpness in dimensions one, two, and three

Page 13 · Corollary 5.2 · arXiv:2608.01443v1

The one-dimensional example is direct, the cited planar sharpness agrees with the earlier theorem, and exact integer arithmetic for $\boldsymbol\alpha=(27,97,514)/1334$ and $N=58$ gives eight record denominators in the required interval. Proposition 5.1 adds the initial value, producing exactly nine distances.

Dettmann, exact three-dimensional example ↗

02Proofs4 reported findingsCorrect

The central proofs and all material internal inputs checked are correct and complete after immediate-consequence closure. One mechanical dimension-index typo remains visible in yellow and does not lower the proof status.

Theorem 1.4 and Corollary 3.3Correct and complete

Sum-free, pairing, and inductive-growth arguments

Pages 5–10 · Section 3 · arXiv:2608.01443v1

Every halved signed denominator in the parity argument is integral and lies in the range required by the record-minimum property. The equality case of the sum-free lemma yields the asserted pairing, the four-sign calculation proves the exact denominator identities, and both branches of the dichotomy give the stated inductive exponential-growth bound.

Theorem 4.1Correct and complete

Strict kissing-number recurrence

Pages 10–11 · Theorem 4.1 · arXiv:2608.01443v1

The record-minimum property and the norm identity force the normalized error vectors to have pairwise inner product strictly below $1/2$ . The resulting strict spherical-code bound supplies the claimed denominator recurrence without an unhandled equality case.

Proposition 5.1 and Theorem 1.1Correct and complete

Nearest-neighbour formula and final counting argument

Pages 11–13 · Section 5 · arXiv:2608.01443v1

Symmetry reduces each orbit point's nonzero displacements to the correct initial denominator range, and the endpoint values cover every record minimum used in the exact count. When the rotation has finite order and $N$ is at least that order, every orbit index sees the same set of nonzero displacement representatives, so the terminal case is complete.

Historical boundTypo

The dimension exponent is printed as the orbit-point index

Page 2 · Introduction, first full paragraph · arXiv:2608.01443v1

The recalled bound is printed as $g_N(\boldsymbol\alpha,\mathcal L)\leq3^n+1$ . Replace $n$ by $d$ , giving $g_N(\boldsymbol\alpha,\mathcal L)\leq3^d+1$ . Here $d$ is the dimension governing the cited bound, whereas $n$ denotes an orbit-point index elsewhere; the following dimension-specific discussion fixes the correction uniquely. It is not used in the paper's proof.

Biringer–Schmidt, dimension-dependent distance bound ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.01443v1
Authors listed: Nikita Shulga
Audit date: August 15, 2026

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Nine-distance theorem and growth of best-approximation denominators

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Growth dichotomy for best-approximation denominators

The 2d+12^d+12d+1 bound and the nine-distance theorem

Sharpness in dimensions one, two, and three

Sum-free, pairing, and inductive-growth arguments

Strict kissing-number recurrence

Nearest-neighbour formula and final counting argument

The dimension exponent is printed as the orbit-point index

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The $2^d+1$ bound and the nine-distance theorem