The Minkowski chain and Diophantine approximation

Nickolas Andersen; William Duke

REVIEW 3 major objections 3 minor 25 references

The Minkowski chain and Diophantine approximation

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Minkowski chain separates badly approximable forms from singular ones, and yields a Dirichlet-type theorem that produces a whole basis of approximating integer vectors.

desk verdict Solid, well-written paper with genuinely new Minkowski-chain criteria for badly approximable and singular forms; the central arguments hold up, but the proofs of Theorems 4 and 5 have localized gaps that need fixing before publication. read the letter →

arxiv 1908.06157 v1 pith:ZYD5JFRV submitted 2019-08-16 math.NT

Nickolas Andersen , William Duke This is my paper · ORCID

classification math.NT MSC 11J1311J7011H06

keywords MinkowskichainHurwitzbadlyapproximableformssingularlinearDirichletapproximationsuccessiveminimareducedbasesLiouvillenumbers

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a classical multidimensional continued-fraction algorithm, the Minkowski chain, exactly detects two contrasting Diophantine behaviours of a real linear form $L_\alpha(x)=\alpha_1x_1+\cdots+\alpha_nx_n$. After pushing $\alpha$ through the chain and reading the first coordinate $\alpha_{k,1}$ of the transformed tuple, the form is badly approximable precisely when $|\alpha_{k,1}|$ stays bounded away from zero, and singular precisely when $|\alpha_{k,1}|\to 0$. The same machinery gives a new variant of Dirichlet's approximation theorem: a linear form is badly approximable if and only if, at every scale $Q$, there is a unimodular integer matrix $A$ with $\|A\|_\infty

What carries the argument

The Minkowski chain is the lexicographic algorithm that, for each $m$, chooses an $\ell\times\ell$ nonsingular integer matrix $A_m$ whose rows minimize $\|A_m(\alpha_1,\dots,\alpha_n,1)^\top\|_\infty$ row by row, taking the subsequence of distinct matrices as $B_k$. Its coordinate $\alpha_{k,1}=\beta_1/\beta_\ell$ is the quantity whose size separates badly approximable from singular behaviour. The proof machinery also uses the one-parameter determinant-one lattice $\Lambda_t\subset\mathbb{R}^\ell$, the sup-norm successive minima $\lambda_1(t)$, and a reduced-basis theorem that bounds the product $\lambda_1\cdots\lambda_\ell$ from both sides. The load-bearing bridge is Lemma 5.3, which converts bad approximability into $|\beta_1|\gg m^{-n}$ and $|\beta_\ell|\ll m^{-n}$; the conversion rests on a volume estimate for the auxiliary body $\{G_m<1\}$, whose volume is claimed to be at least $V|\beta_\ell|/m$.

What would settle it

Run the Minkowski chain algorithm on a concrete badly approximable form, for instance $\alpha=(\theta^2,\theta)$ with $\theta=2\cos(2\pi/7)$, and compute $|\alpha_{k,1}|$ for large $k$: Theorem 3 predicts a positive lower bound (indeed the tuples are finite), so any computed value below that bound contradicts the theorem. Conversely, for a Liouville number such as $\lambda=\sum_{m\ge1}10^{-m!}$ and a fixed $n$, Theorems 3 and 5 predict $|\alpha_{k,1}|\to0$; a direct chain computation that stabilizes away from zero would refute the claimed dichotomy.

Watch

Extended reading notes

Core claim

At the centre is Theorem 6, with Theorems 3 and 5 giving the two halves of the same dichotomy. For $\alpha=(\alpha_1,\dots,\alpha_n)$ with $1,\alpha_1,\dots,\alpha_n$ linearly independent over $\mathbb{Q}$, the Minkowski chain produces matrices $B_k$ and tuples $B_k(\alpha)=(\alpha_{k,1},\dots,\alpha_{k,n})$ with $0<|\alpha_{k,1}|<\cdots<|\alpha_{k,n}|<1$. The paper proves that $L_\alpha$ is badly approximable exactly when $\inf_k|\alpha_{k,1}|>0$, and singular exactly when $\lim_{k\to\infty}|\alpha_{k,1}|=0$. The proof links the chain coefficients $\beta_1,\dots,\beta_\ell$ (with $\ell=n+1$) to the successive minima of the lattice $\Lambda_t$ of determinant one: bad approximability forces the power-law spacing $cm^{-n}<|\beta_1|<\cdots<|\beta_\ell|<Cm^{-n}$, while failure of bad approximability makes $|\beta_1/\beta_\ell|=|\alpha_{k,1}|$ arbitrarily small for some $k$. Theorem 6 upgrades the usual one-vector Dirichlet bound to a full basis statement, and its converse shows that, when the form is not badly approximable, every unimodular matrix with $\|A\|_\infty<Q$ has row error larger than $cQ^{-n}$. For $n=1$ this recovers the classical fact that an irrational is badly approximable iff its partial quotients are bounded.

Load-bearing premise

The proof depends on a stated but deferred volume calculation for the auxiliary norm body $\{G_m<1\}$; every later inequality inherits its constants from that estimate, so if the true volume is smaller than claimed, the lattice argument would not establish the characterization.

Editorial extensions

If this is right

For $n=1$, Theorem 3 together with the chain-to-continued-fraction dictionary recovers the classical fact that an irrational is badly approximable iff its partial quotients are bounded.
For every Liouville number $\alpha$ and every fixed $n$, the form $L_{(\alpha^n,\dots,\alpha)}$ is not badly approximable, so its Minkowski-chain coordinate $|\alpha_{k,1}|$ tends to zero as $k\to\infty$.
Whenever $L_\alpha$ is badly approximable, Corollary 3 produces infinitely many unimodular matrices $A\in\mathrm{GL}(\ell,\mathbb{Z})$ with $\|A(\alpha_1,\dots,\alpha_n,1)^\top\|_\infty<c\|A\|_\infty^{-n}$.
Minkowski's original finite-set criterion for algebraic numbers is proved with the same machinery: $\alpha$ is algebraic of degree $\ell$ exactly when the chain tuples form a finite set.
The lattice form of the criterion connects the algebraic chain to bounded trajectories in the space of unimodular lattices, giving a geometric certificate for bad approximability and singularity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same reduced-basis argument should extend to systems of several linear forms, since the proof never uses special features of a single form beyond the lattice $\Lambda_t$; the paper itself notes this as a natural next step.
Because the Minkowski chain is an explicit algorithm, Theorem 3 suggests a numerical test: truncate the chain and monitor $|\alpha_{k,1}|$; a visible positive floor would be evidence of bad approximability, whereas repeated dips toward zero would indicate singularity.
Theorem 6 can be read as saying that bad approximability is equivalent to a uniform basis-valued Dirichlet condition; an interesting open question is the optimal size of the constant $c$ and whether the full-basis conclusion can be quantified effectively from a finite prefix of the chain.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth engaging. The new results are real: Theorems 3, 5, and 6 give Minkowski-chain criteria for badly approximable and singular forms and a genuine basis-level Dirichlet variant. The paper is also a useful revival of Minkowski's old algorithm, with a self-contained treatment and tracked constants. The central arguments for Theorems 3 and 6 look sound to me; the lattice machinery (reduced bases, First Finiteness Theorem) is applied coherently.

The soft spots are all in the proofs of Theorems 4 and 5. Theorem 4's proof is wrong as written: it claims that for a Liouville α, for all x with 0<||x||∞≤Q, ||L(α^n,...,α)(x)|| ≥ (1−ε)Q^{−n}. But for x=(0,...,0,1), the form value is α itself, which can be much smaller than Q^{−n} because α is Liouville. That stronger claim is false. The theorem itself is still true, since the single vector (0,...,0,1) already violates bad approximability, but the proof needs to be replaced, not patched.

Theorem 5 has two localized issues. First, the negation of 'singular' is misstated: it says there are infinitely many q with ||L(q)|| ≥ c Q^{−n}, but the later minimum bound requires the inequality to hold for all nonzero q up to Q. Second, in the singular→|α_{k,1}|→0 direction, the text sets t=m/ε and then treats a lattice vector as admissible for the Minkowski index m without ensuring |p|≤m; the missing κ factor from Theorem 3 is exactly what is needed. The stress-test note on this is correct. Both are easily repairable.

So the paper is not in publishable condition as-is, but the flaws are not fatal to the main claims. The novelty and the central criteria justify sending it to a careful referee, who should ask for corrections to these two proofs. I would cite the paper once the corrections are in.

Referee Report

3 major / 3 minor

Summary. The paper studies the Minkowski chain, a multidimensional generalization of the Hurwitz continued fraction chain, and applies it to Diophantine approximation. The authors prove that a real linear form L_α is badly approximable if and only if the first coordinate of the Minkowski chain is bounded away from zero (Theorem 3), and singular if and only if that coordinate tends to zero (Theorem 5). They also prove a Dirichlet-type theorem (Theorem 6) asserting that L_α is badly approximable exactly when for every Q there is a unimodular integral matrix with entries below Q whose image under the vector (α,1) has sup norm below c Q^{-n}. Along the way they give a proof of Minkowski's algebraic criterion (Theorem 2) and a consequence for Liouville numbers (Theorem 4). The proofs use Minkowski's second theorem on successive minima and a theory of reduced bases, with constants tracked explicitly in the central lemmas.

Significance. If the results are correct, the paper provides explicit and elegant arithmetic criteria for bad approximability and singularity, and a new sharp form of Dirichlet's theorem with a full basis of approximating vectors. The use of the Minkowski chain revives a classical algorithm and connects it to modern Diophantine approximation and to Dani's dynamical criteria. The proofs are constructive, and the reduction to the lattice Λ_t via successive minima and reduced bases is a clean and potentially influential technique. The paper also gives a self-contained treatment of the necessary geometry-of-numbers background, including a proof of the First Finiteness Theorem. These strengths make the paper a valuable contribution to the field, provided the localized gaps in the proofs of Theorems 5 and 4 are repaired as indicated below.

major comments (3)

[§7, proof of Theorem 5, first paragraph] In the direction 'singular ⇒ |α_{k,1}|→0', the proof sets t = m/ε and obtains from λ1(t) < ε an integral vector r with |q_j| < εt = m for j = 1,...,n and |ξ(r)| < ε t^{-n}. It then claims |β1| < ε/t^n by comparing with r. However, for r to be admissible for the Minkowski algorithm at level m one needs ‖r‖∞ ≤ m. While |q_j| ≤ m is automatic, the bound on the last coordinate p is |p| ≤ |ξ(r)| + Σ |α_j| |q_j| < ε t^{-n} + (Σ |α_j|) m, which exceeds m when Σ |α_j| > 1. Thus r need not be admissible, and the inequality |β1| ≤ |ξ(r)| is not justified. This is precisely the issue that is correctly handled in the proof of Theorem 3 by taking m = ⌈κ t ε⌉ with a constant κ depending on α. The proof of Theorem 5 should incorporate the same κ; after that change the argument goes through with the same ε-dependence up to constants. Please repair this step.
[§7, proof of Theorem 5, converse] The negation of singularity is misstated. The text says that if α is not singular, 'there exists a c > 0 and a sequence {Q_j} tending to infinity such that for each j there are infinitely many q ∈ Z^n with ‖q‖∞ ≤ Q_j and ‖L_α(q)‖ ≥ c Q_j^{-n}.' This is not equivalent to the negation of the definition of singularity; it is much weaker, and the phrase 'infinitely many q' is neither necessary nor sufficient for the subsequent lower bound on β1. The correct negation is: there exists c > 0 and a sequence Q_j → ∞ such that for all q with 0 < ‖q‖∞ ≤ Q_j one has ‖L_α(q)‖ ≥ c Q_j^{-n}. With this correction, the rest of the proof, which follows the lines of Lemma 5.3, is sound.
[§5, Lemma 5.3] The proof relies on the assertion 'A straightforward calculation shows that vol({x; G_m(x) < 1}) ≥ V |β_l|/m.' This volume bound is the only quantitative bridge between the Minkowski chain coefficients and the lattice Λ_t, and all constants in Lemma 5.3, and hence the quantitative forms of Theorems 3 and 5, depend on it. Since the paper promises proofs of all numbered lemmas, please supply the calculation or a precise reference. The bound is plausible (the body contains a slab of width proportional to |β_l|/m in the last coordinate for a positive-measure set of the first n coordinates), but the explicit dependence of V on α should be recorded.

minor comments (3)

[§7, proof of Theorem 4] The assertion 'x_1 p^n + ... + x_n p q^{n-1} - y q^n is a non-zero integer since q ∤ x_1' fails when x_1 = 0. The conclusion of the lower bound remains true: if x_1 = 0, let j be the smallest index with x_j ≠ 0; after dividing by q^{j-1}, the expression is congruent to x_j p^{n-j+1} modulo q and hence is nonzero, giving an even stronger lower bound. Please adjust the argument to cover this case.
[§2, around (2.4)] The strict inequalities 0 < |α_{k,1}| < ... < |α_{k,n}| < 1 are asserted without proof. The sequential minimization gives |β_1| ≤ ... ≤ |β_l|, but strictness is not automatic; ties could occur even under the Q-linear independence assumption (for instance, two independent minimizers with opposite signs). Please either prove the strict inequalities or weaken them to non-strict where only non-strict inequalities are needed in the subsequent arguments.
[Throughout] There are several typographical errors: 'Fibonnaci' in Example (i), 'Minkowsi' in the Introduction, and garbled umlauts in the reference list (e.g., 'Bercksichtigung' for 'Berücksichtigung'). These should be corrected in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Minkowski-chain criteria are proven from geometry-of-numbers and lattice arguments, with no fitted inputs or self-citation load-bearing steps.

full rationale

The paper's central claims (Theorems 3, 5, and 6) are derived self-containedly. The Minkowski chain is defined by an independent minimization procedure over rows of integral matrices, while badly approximable and singular are defined by the standard inequalities on the linear form. The bridge between them is supplied by explicit lattice constructions, successive-minima theorems, and reduced-basis inequalities, all treated as external mathematical input and either proved in the paper or cited to sources outside the authors' own prior work. Lemma 5.3 converts bad approximability into bounds on the chain coefficients using Minkowski's successive-minima theorem with tracked constants; no parameter is fitted to the predicted quantity. The proof of Minkowski's criterion (Theorem 2) is also provided in Section 5, so the attribution to Minkowski is not an unexamined self-citation. The possible gap in the singular direction of Theorem 5, concerning the coefficient-dependent scaling needed for admissibility of the lattice vector, is a localized proof-repair issue rather than a circularity: it does not make the conclusion equivalent to an assumption. No quantity is fitted to data, and no result is assumed in the form it claims to prove.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; all constants c, C, V, κ are introduced as abstract dependencies on α and are never assigned numerical values. The axioms are standard results in geometry of numbers and Diophantine approximation plus the explicit linear-independence domain assumption. No new entities such as particles, forces, or dimensions are postulated.

assumptions (5)

domain assumption The set {α1,...,αn,1} is linearly independent over Q.
Stated in §2 and in Theorems 3 and 5. It guarantees uniqueness of the Minkowski chain; without it the lexicographic minimization is not well-defined.
standard math Minkowski's theorem on successive minima for determinant-one lattices.
Invoked in §5 to derive the two-sided bounds on β1 and βl in Lemma 5.3; cited from [19].
standard math A Q-basis of a real number field gives a badly approximable linear form (Schmidt, Diophantine Approximation, Thm 4A).
Used in the proof of Theorem 2 to obtain the lower bound (3.1).
standard math Davenport-Schmidt theorem: if a form is badly approximable, Dirichlet's theorem can be improved with a constant c<1.
Used in Theorem 4's intended contrapositive argument; cited from [7].
standard math The sup-norm unit ball in R^l has volume 2^l.
Implicit in applications of the First Finiteness Theorem and Lemma 7.1.

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Cite this review

Pith. "Pith review of The Minkowski chain and Diophantine approximation." pith.science (2026). https://pith.science/paper/ZYD5JFRV

@misc{pith2026190806157,
  author       = {Pith},
  title        = {Pith review of: The Minkowski chain and Diophantine approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZYD5JFRV}},
  note         = {Machine review of arXiv:1908.06157}
}

read the original abstract

The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's generalization (the Minkowski chain) to give criteria for a real linear form to be either badly approximable or singular. We also give a variant of Dirichlet's approximation theorem for a real linear form that produces a whole basis of approximating integral vectors rather than a single one. This result holds if and only if the form is badly approximable. The proofs rely on properties of successive minima and reduced bases of lattices.

Figures

Figures reproduced from arXiv: 1908.06157 by the authors.

**Figure 1.** The sequences |λk,1| for n = 1, 2, 3. which corresponds to (2.1). Here Fk is the k th Fibonnaci number and for each k Bk( −1+√ 5 2 ) = 1− √ 5 2 . (ii) Let θ = 2 cos 2π 7 so that Q(θ) is the real cubic field of discriminant 49, i.e. the splitting field of x 3 + x 2 − 2x − 1. The Minkowski chain for (θ 2 , θ) begins B1 = 0 1 −1 1 −1 0 1 −1 −1 , B2 = 1 −2 1 2 −1 −2 0 1 −1 , B3 = 1 −2 1 3 −3 −1 2 0 −3 , B4… view at source ↗

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Cassels, J. W. S., An introduction to the geometry of numbers. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Bercksichtigung der Anwendungsgebiete, Bd. 99 Springer-Verlag, Berlin-Gttingen-Heidelberg 1959 viii+344 pp

work page 1959

[2] [2]

Cheung, Y., Hausdorﬀ dimension of the set of singular pairs. Ann. of Math. (2) 173 (2011), 127167

work page 2011

[3] [3]

& Chevallier, N., Hausdorﬀ dimension of singular vectors

Cheung, Y. & Chevallier, N., Hausdorﬀ dimension of singular vectors. Duke Math. J. 165 (2016), 22732329

work page 2016

[4] [4]

G., Divergent trajectories of ﬂows on homogeneous spaces and Diophantine approximation

Dani, S. G., Divergent trajectories of ﬂows on homogeneous spaces and Diophantine approximation. J. Reine Angew. Math. 359 (1985), 5589

work page 1985

[5] [5]

& Urbaski, M., A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation

Das, T., Fishman, L., Simmons, D. & Urbaski, M., A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation. C. R. Math. Acad. Sci. Paris 355 (2017), 835846

work page 2017

[6] [6]

Quarterly J

Davenport, H., Minkowski’s inequality for the minima associated with a convex body. Quarterly J. of Math., Volume os-10, Issue 1, (1939), 119121

work page 1939

[7] [7]

& Schmidt, W

Davenport, H. & Schmidt, W. M., Dirichlet’s theorem on Diophantine approximation. 1970 Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) pp. 113132 Academic Press, London

work page 1970

[8] [8]

Gruber, P. M. & Lekkerkerker, C. G., Geometry of numbers. Second edition. North-Holland Mathematical Library, 37. North-Holland Publishing Co., Amsterdam, 1987. xvi+732 pp

work page 1987