Infinite rational distance sets in affine general position: constructions in every dimension

Junkai Qiu

REVIEW 2 major objections 4 minor 28 references

Infinite rational distance sets in affine general position: constructions in every dimension

T0 review · 2 major / 4 minor · reviewed 2026-08-28 · deepseek-v4-flash

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Pith's one-line read For every dimension d, there is a countably infinite set of points in R^d with rational pairwise distances and with every d+1 points affinely independent; for odd d, no d+2 points are cospherical.

desk verdict Genuinely new construction with a coherent core; verify the Browning–Matthiessen application in Lemma 4.5 and the ancillary computation before relying on the d ≡ 2 (mod 4) rational-coordinate claim. read the letter →

arxiv 2608.23529 v1 pith:PWR4CHPO submitted 2026-08-24 math.CO math.MG

Junkai Qiu This is my paper · ORCID

classification math.COmath.MG MSC 52C1051K0511E1252B11

keywords rationaldistancesetsintegralpointgeneralpositionChebyshevpolynomialsquadraticformscyclicpolytopesaffineindependencestereographicprojection

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

This paper tries to establish that Euclidean space of every dimension $d\geq 1$ contains a countably infinite set of points whose pairwise distances are all rational and which is as nondegenerate as possible: every $d+1$ distinct points are affinely independent, and when $d$ is odd no $d+2$ points lie on a common sphere. The proof is uniform in $d$: rational parameters on a short arc of a harmonic curve built from Chebyshev polynomials give points whose chord lengths are rational because of a squared polynomial identity. A divided-difference Wronskian argument shows the short arc is locally convex, so affine independence is automatic. The same construction yields infinite rational-coordinate sets in $\mathbb{Q}^d$ for every even $d$ and for $d\equiv 1\pmod 4$, finite integral point sets realizing all cyclic polytopes, and infinitely many primitive three-dimensional integral clusters.

What carries the argument

The load-bearing object is a harmonic curve $\Gamma_r$ built from even-indexed Chebyshev polynomials of the second kind; the product identity $U_{2i}U_{2j}=U_{i+j}^2-U_{j-i-1}^2$ makes the squared chord length equal to $4s^2P_r(c)^2$, hence rational for rational circle parameters. The companion mechanism is a divided-difference factorization of the affine determinant: a nonzero Wronskian $\det[\gamma'(0),\ldots,\gamma^{(n)}(0)]$ on a short interval forces every $n+1$ distinct points on the arc to be affinely independent, and for this curve the Wronskian factors into products of pairwise differences of the distinct squared frequencies. For the rational-coordinate theorems, the additional mechanism is a positive-definite rational quadratic form with paired diagonal blocks that is proved, through local norm-residue invariants and the local-global classification of rational quadratic forms, to be rationally isometric to the standard sum-of-squares form.

What would settle it

Evaluate the explicit three-dimensional distance formula (48) at two rational parameters in $(0,1/6)$; any irrational value would falsify the rational-distance claim. More globally, check the asserted nonzero determinant over $\mathbb{F}_{23}$ in the nine-point example of Proposition 6.2: a vanishing determinant would mean the nine points are not affinely independent, destroying that example.

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Extended reading notes

Core claim

The central claim is a single infinite family of curves, one for each dimension, that settles the existence of infinite rational distance sets under strong nondegeneracy. In even dimension $d=2r$, the set $X_{2r}$ is the image, under a rational parametrization of the unit circle, of a short arc of the curve $\Gamma_r(z)=(\sqrt{\lambda_k}\,\Re z^{2(k+1)},\sqrt{\lambda_k}\,\Im z^{2(k+1)})_{k\in K_r}$, and the identity $\|\Gamma_r(z)-\Gamma_r(w)\|=2|sP_r(c)|$, with $z\bar w=c+is$, makes every rational-parameter chord rational. The Wronskian of this curve factors into products of pairwise differences of the distinct frequencies, so every $2r+1$ points on a sufficiently short arc are affinely independent. For odd $d=2r-1$, stereographic projection from the endpoint $N=\Gamma_r(1)$ maps the arc into the hyperplane $N^\perp$, preserves rational distances, and converts any affine dependence or cosphericality among projected points into a hyperplane section of the source arc, which local convexity forbids. The paper also proves that replacing the circle by a rational norm conic and identifying the resulting quadratic form with the standard Euclidean form places infinitely many points in $\mathbb{Q}^d$ for even $d$ and for $d\equiv 1\pmod 4$.

Load-bearing premise

The least secure load-bearing premise is the application of a deep norm-form density theorem to the specific linear forms arising in the rational-coordinate construction for even dimensions that are two more than a multiple of four; if that application fails, those dimensions lose their proof, while the main real-space construction and the other rational-coordinate families stand.

Editorial extensions

If this is right

For every $d\geq 1$ there is a countably infinite rational distance set in $\mathbb{R}^d$ in affine general position, and its cardinality is optimal since any full-dimensional rational distance set injects into $\mathbb{Q}^{d+1}$.
For odd $d$, the set can be chosen in full general position, so in dimension three there is an infinite rational distance set in which every four points are noncoplanar and no five points are cospherical.
For every even $d$ and every $d\equiv 1\pmod 4$, the same construction gives infinite rational distance sets inside $\mathbb{Q}^d$, and finite subsets rescale to lattice integral point sets.
Taking and rescaling $n$ suitable points gives $n$-point integral point sets with the alternating order type, so their convex hulls realize the cyclic polytope $C(n,d)$ for all $d\geq 2$ and $n\geq d+1$.
In dimension three, the explicit rational parametrization yields infinitely many pairwise non-similar primitive $n_3$-clusters for every $n\geq 4$.

Reading between the lines

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

The paper's own limitation remarks suggest that the even-dimensional construction is inherently spherical, so full general position in even dimensions would require a genuinely different source curve; the paper leaves that as an open route rather than a claim.
Because the rational-coordinate branch for dimensions two more than a multiple of four depends on a deep, non-effective density theorem, one straightforward test is to try to replace it in a fixed small dimension by an elementary norm search; success would make those $\mathbb{Q}^d$ constructions unconditional and effective.
The nine-point characteristic-one set that admits no rational realization is a proof of concept that the common characteristic is not a sufficient rational-realizability invariant in high dimensions; searching for the minimal dimension of such an example is a natural next step, since the paper does not claim eight is minimal.
The explicit three-dimensional curve could be optimized directly: choosing rational parameters $u_j=1/M_j$ gives crude effective diameter bounds, and a systematic search over rational parameters might produce substantially better primitive cluster diameters than the paper records.

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Formalized claims in Lean

Claim #1: The central claim is a single infinite family of curves, one for each dimension, that settles the existence of infinite rational distance sets under strong nondegeneracy. In even dimension $d=2r$, the set $X_{2r}$ is the image, under a rational parametrization of the unit circle, of a short arc of the curve $\Gamma_r(z)=(\sqrt{\lambda_k}\,\Re z^{2(k+1)},\sqrt{\lambda_k}\,\Im z^{2(k+1)})_{k\in K_r}

/-- @claim 1 The central claim is a single infinite family of curves, one for each dimension, that settles the existence of infinite rational distance sets under strong nondegeneracy. In even dimension $d=2r$, the set $X_{2r}$ is the image, under a rational parametrization of the unit circle, of a short arc of the curve $\Gamma_r(z)=(\sqrt{\lambda_k}\,\Re z^{2(k+1)},\sqrt{\lambda_k}\,\Im z^{2(k+1)})_{k\in K_r} -/
def central_claim : Prop :=

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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

Briefly: this is a genuinely new construction paper, and the core real-coordinate part holds up on my reading. The main thing to check before relying on it is the compressed application of Browning–Matthiessen in Lemma 4.5, which carries only the d ≡ 2 (mod 4) rational-coordinate case, not the paper's central existence theorem.

The genuinely new ingredients are the Chebyshev square decompositions with rational chord lengths, the Vandermonde-factorized Wronskian showing local convexity, and the stereographic projection step for odd dimensions. Theorems 1.2 and 1.3 (the latter minus the fragile case), Theorem 1.6, and the explicit 3D rational curve of Theorem 5.1 all look like real advances. I checked the product linearization (1), the weight recursion (8), the base identity (10), and the determinant bound (49); they are consistent. The paper is also honest: Remark 4.6 concedes non-effectivity, Remark 4.10 spells out exactly which results are independent of BM17, and Section 6.2 states clearly what the method does not do. Proposition 6.2 is a useful correction of an overstatement in prior literature, and the failure of characteristic 1 to imply rational realizability is a clean example.

The soft spot is real, though narrower than a casual reader might think. Lemma 4.5 invokes Browning–Matthiessen's norm-form theorem with every linear form assigned the same field Q(i). The proof asserts that pairwise nonproportionality is the only nondegeneracy needed, and constructs local points via v_p = p^{2N_p}. The local-adic part is plausible, but the hypothesis check is two sentences, and the theorem is deep enough that a referee should verify this once. If the application fails, the Q^d claim for d ≡ 2 (mod 4), d ≥ 6, loses its proof; the R^d theorems, the d ≡ 0 (mod 4) rational-coordinate case, and the d ≡ 1 (mod 4) general-position case all stand independently, as Remark 4.10 states. That is my reading too.

Minor concern: the numerical verification supporting Proposition 6.2 and the primitive 8_3-cluster is in ancillary files, not auditable from the printed text. For a paper whose main claims are algebraic and exact, this is secondary, but the author should supply the verification inline or make the code citable.

Citation pattern is clean: no self-citations, and the prior literature (Steiger, Anil Kumar, Avdeev) is correctly described as weaker or different. No fitted parameters and no circularity that I could find.

Who should read it: people working on rational distance sets, integral point sets, and maybe moment curves. It deserves a serious referee, and I'd recommend accepting it subject to verification of the BM17 application and of the ancillary computations. Beyond that, the main construction is in good shape.

Referee Report

2 major / 4 minor

Summary. The paper constructs, for every d≥1, a countably infinite set X_d⊂R^d with pairwise rational distances and with every d+1 distinct points affinely independent; for odd d the construction additionally avoids d+2 cospherical points. The method is uniform: a Chebyshev product identity produces a positive linear combination of squares of Chebyshev polynomials, whose square is itself a square of a rational polynomial; this yields a harmonic curve on a sphere whose chords between rational-parameter points have rational lengths. A divided-difference/Wronskian argument proves local affine independence on short arcs, and stereographic projection gives the odd-dimensional examples. The paper also proves rational-coordinate versions in Q^d for even d and for d≡1 mod 4, finite integral point set consequences, integral-distance cyclic polytopes, and a three-dimensional explicit curve giving infinitely many primitive n_3-clusters.

Significance. If correct, the paper resolves the existence question for infinite rational distance sets in affine general position in every dimension, strengthens prior finite and infinite constructions, and gives the first such examples with optimal countable cardinality. The main real-coordinate construction is explicit and parameter-free: the algebraic identities (1), (8), (10), and (11) are checkable, and the proof of Theorem 1.2 is essentially self-contained. The finite consequences for integral point sets and cyclic polytopes are natural and the three-dimensional explicit parametrization is valuable. The weaker point is the rational-coordinate theorem for even dimensions d≡2 mod 4, which depends on a deep external theorem through a compressed hypothesis check, and the paper itself concedes non-effectivity and states clear limitations in Remarks 4.6 and 4.10 and Section 6.2.

major comments (2)

[Section 4.4, Lemma 4.5 (Eq. (30))] The application of Browning–Matthiessen [BM17, Theorem 1.3] is the least secure load-bearing step. The proof reduces the nondegeneracy hypotheses to pairwise nonproportionality of the linear forms and asserts that repeated use of the same norm-field Q(i) is allowed, but this is compressed into two sentences. Theorem 1.3(i) for d≡2 mod 4 with d≥6 depends on this lemma, so the hypothesis check must be fully verifiable. Please quote the exact hypotheses of [BM17, Theorem 1.3] and verify them item by item (number of base variables, multiplicity of the norm form field, nondegeneracy/smoothness, and the required adelic integrability of the local points). If the verification cannot be completed, restrict Theorem 1.3(i) to the cases d≡0 mod 4 and d=2, and adjust the abstract and Remark 4.10 accordingly.
[Section 4.4, Claim 1 in Proposition 4.7] The parity-adjustment argument for v2(A) is load-bearing for the case r≡2 mod 4, but the 2-adic induction is only sketched. After the adjustment b1(δ) is a rational function of δ, and the old weights λk(δ) are also rational functions of δ; the statement that 'the ultrametric inequality leaves all old valuations unchanged' requires a threshold N0 such that for δ=2^N/L with N>N0 every perturbation term has strictly larger 2-adic valuation than the corresponding old term. Please provide the explicit valuation table for all λk(δ) in the induction step, or otherwise justify the preservation of positivity and valuations. Without this, Proposition 4.7 does not fully support the d≡2 mod 4 rational-coordinate claim.

minor comments (4)

[Footnotes 1–2 and Proposition 6.2] The proof of Proposition 6.2 and the illustrative primitive 8_3-cluster rely on exact calculations whose outcomes are stated but not shown in the printed text (e.g., Eq. (52) and the determinant bound in the cluster verification). Please include the relevant data or a precise reference to a permanent supplementary file with a checksum or version identifier, so that the computations are auditable without external files.
[Lemma 2.1 and Eq. (4)] The definition of K_r is somewhat confusing because K_r={0,...,r-1} for odd r and K_r={1,...,r} for even r; while both have cardinality r, the shift is easy to miss and affects the frequencies 2(k+1). A short comment that the indexing convention is chosen to make the base cases of the positivity recursion clean would help the reader.
[Section 5.2, proof of Theorem 5.1] The bound 0<y_j≤tan(φ(1/6)/2)=12√5/31<1 is used to conclude all elementary symmetric polynomials e_k are positive; it would be helpful to include the one-line derivation of the value tan(φ(1/6)/2) from the parametrization (46), since it is not immediate.
[Section 6.2] In the discussion of d≡3 mod 4, the statement that the remaining conditions 'form a simultaneous Hilbert-symbol selection problem' is clear but a small worked example for one d≡3 mod 4 value would make the obstruction and the scope of the method more transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Chebyshev-square construction is target-free and its load-bearing inputs are explicit algebraic identities or independent external theorems.

full rationale

I walked the derivation chain and found no step in which a claimed output is equivalent to its input by construction. Lemma 2.1 builds positive rational weights and a polynomial P_r satisfying P_r^2 = Sum lambda_k U_k^2 by an explicit inductive perturbation argument; no desired distance set is used to choose the weights. The chord-length formula (6) then follows directly from the Chebyshev product identity, and the affine-independence criterion follows from the divided-difference factorization of Lemma 2.2 and the Wronskian evaluation (11), both proved in the paper. The stereographic projections in Propositions 3.2 and Lemma 4.4 are exact algebraic identities, and the odd-dimensional cosphericity argument reduces to a proper affine hyperplane equation derived in the text. The rational-coordinate section uses external machinery, namely the Hasse-Minkowski classification of quadratic forms (Lemma 4.3, via Serre) and the Browning-Matthiessen norm-form theorem (Lemma 4.5), neither of which is authored or presupposed by this paper. The caveats in Remark 4.6 (non-effectivity of the norm choice) and Remark 4.10 (the theorem is used only for d congruent 2 mod 4 rational-coordinate refinements) are limitations on effectivity and scope, not hidden assumptions of the main theorems. There are no self-citations by the author, and Proposition 6.2 is a self-contained counterexample with computed Hilbert symbol rather than an imported claim. Finite consequences and cyclic polytopes are obtained by taking finite subsets, ordered parameters, and scaling, with no parameter fitted to a target configuration. I therefore find no reduction of any claimed result to its own inputs.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

All input quantities are constructed within the paper or taken from standard published theorems; no parameter is fitted to any target data set. The free parameters listed are existence constants from continuity or weak-approximation arguments, not fitted values. The invented-entities list is empty: the paper postulates no new objects beyond the constructions themselves.

free parameters (4)

delta_r (short-arc length) = none, existential
Lemma 2.2 guarantees some delta > 0 with nonzero divided-difference quotient on [0,delta]^(n+1); the construction needs no explicit value.
eta_r (parameter cutoff) = none, chosen so 2 arctan(eta_r) < delta_r
Defines X_2r and X_(2r-1) as rational parameters in (0,eta_r); any sufficiently small positive rational works.
epsilon (induction perturbation) = none, any sufficiently small positive rational
Lemma 2.1 builds positive weights by appending b_m = epsilon and relying on continuity of the weights at epsilon = 0; the choice is existential.
t (weak-approximation parameter) = none, non-effective
Lemma 4.5 supplies rational t with t, 1+alpha_j t Gaussian norms via weak approximation; no bound is given, and the paper concedes this in Remark 4.6.

assumptions (8)

standard math Chebyshev product linearization U_(2i)U_(2j) = U_(i+j)^2 - U_(j-i-1)^2
Proved in Section 2 by trigonometric telescoping; used in Lemma 2.1 to express squares of linear combinations of even-indexed polynomials.
standard math Divided-difference determinant factorization and Wronskian continuity (de Boor)
Lemma 2.2 relies on the standard divided-difference factorization and repeated-node limits; the nonzero Wronskian at 0 yields affine independence on a short arc.
standard math Hasse-Minkowski classification of rational quadratic forms (dimension, determinant square class, Hasse invariants)
Lemma 4.3, citing Serre, is the backbone of the rational-coordinate theorems: it decides when the constructed form is equivalent to the standard sum of squares.
standard math Hasse norm theorem for quadratic extensions
Used in (25) to translate simultaneous norm conditions into Hilbert symbol equalities for the d ≡ 1 (mod 4) case and the Gaussian norm conditions.
standard math Gauss-Legendre three-square theorem
Used in Propositions 4.7 and 4.9 to write A as a sum of three squares when v_2(A) is odd, and again for the k-splitting.
standard math Browning-Matthiessen theorem on norm forms as products of linear polynomials [BM17, Thm 1.3]
The deep external input in Lemma 4.5; it supplies rational points with simultaneous Gaussian norm conditions. Load-bearing only for d ≡ 2 (mod 4), d ≥ 6, as Remark 4.10 states.
standard math Marshall-Perlis rational-to-lattice embedding in dimension three [MP13, Thm 6]
Used in the proof of Theorem 1.7 to rotate a primitive integral rational-coordinate set into Z^3; the paper notes no all-dimensions analogue is known.
standard math Rational two-square theorem and norm criterion for Q(i)
Used in Remark 4.6 for effectivity of the Gaussian norm search and in the Hilbert symbol computations.

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Cite this review

Pith. "Pith review of Infinite rational distance sets in affine general position: constructions in every dimension." pith.science (2026). https://pith.science/paper/PWR4CHPO

@misc{pith2026260823529,
  author       = {Pith},
  title        = {Pith review of: Infinite rational distance sets in affine general position: constructions in every dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PWR4CHPO}},
  note         = {Machine review of arXiv:2608.23529}
}

abstract

For every integer $d\geq 1$, we construct a countably infinite set $X_d\subset\mathbb{R}^d$ in affine general position, with all pairwise distances rational. When $d$ is odd, $X_d$ may also be chosen so that no $d+2$ points lie on a common sphere. The construction is uniform in $d$: positive Chebyshev square decompositions produce harmonic curves on spheres whose points corresponding to rational parameter values have pairwise rational distances. A divided-difference factorization of the affine determinant shows that sufficiently short arcs are locally convex, and stereographic projection produces the odd-dimensional examples. We also construct infinite rational distance sets in $\mathbb{Q}^d$ in affine general position for every even $d$, and in general position for every $d\equiv 1\pmod 4$. For every $d\geq 1$ and $n\geq d+1$, taking and rescaling suitable finite subsets gives $n$-point integral point sets in affine general position. A suitable ordered choice yields integral-distance realizations of all cyclic polytopes. In dimension three, we give an explicit rational parametrization and obtain infinitely many pairwise non-similar primitive $n_3$-clusters for every $n\geq 4$.

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Reference graph

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Pith tools

Reviewed August 28, 2026 · model on record in the stance chip above.

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