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On the probability of n equidistant points in high-dimensional lattices

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Pith's one-line read In d dimensions, the chance that n iid lattice vectors are pairwise equidistant decays as an explicit constant times a power of 1/d, with the exponent fixed by n.

desk verdict Theorem 3.3 as stated is not scale-invariant and fails for X uniform on {0,2}, but the proof gives a sound method for a normalized version; the paper deserves a serious referee. read the letter →

arxiv 2502.03440 v1 pith:P33LUMYD submitted 2025-02-05 math.PR

Stefan Gerdjikov , Martin Minchev , Mladen Savov This is my paper · ORCID
classification math.PR MSC 60F0560G5005C5011H06
keywords MultidimensionallocallimittheoremDiscretelatticeLinegraphsSpectralgraphtheoryEquidistantpointsbinarycodesdistributionsJohnson
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

Let $X_1,\dots,X_n$ be independent $d$-dimensional vectors whose entries are iid samples from a finite lattice distribution $X$, and let $p_d$ be the probability that all their pairwise Euclidean distances are equal. This paper proves that as $d \to \infty$, $p_d \sim C_n / d^{(m-1)/2}$ with $m = \binom{n}{2}$, where the constant $C_n$ is given explicitly in terms of the first four moments of $X$ (Theorem 3.3). The same method also handles the symmetric Bernoulli case, all finitely supported distributions (Theorem 3.8), and other distances such as $L^p$ distances. The result gives the asymptotic density of equidistant families in high-dimensional discrete spaces, connecting a probabilistic question to equidistant binary codes and to the geometry of lattices.

What carries the argument

The mechanism has three parts. First, the multidimensional local limit theorem for lattice distributions converts the probability into $| \operatorname{Lat} V | (2\pi d)^{-(m-1)/2} |\operatorname{Var}(V)|^{-1/2}$ for the increment vector $V = ((X_i-X_j)^2-(X_1-X_2)^2)_{(i,j)\ne(1,2)}$. Second, the paper analyzes the image of the 'overlapping map' $H:\{0,1\}^n \to \{0,1\}^{\binom{n}{2}}$, $(v_1,\dots,v_n)\mapsto(\mathbf{1}_{v_i\ne v_j})_{i<j}$; Theorem 2.1 gives a basis of this lattice consisting of $H([1:i])$ for $i<n$ together with all $2e_{i,j}$ with $j<n$, and after removing the $(1,2)$ coordinate this yields a universal basis with fundamental volume $2^{m-n}$. Third, the covariance determinant is computed by recognizing the $m\times m$ matrix of overlap counts as the adjacency matrix of the line graph $L(K_n)$ (the triangular graph), whose eigenvalues are $2n-4$, $n-4$, and $-2$ with multiplicities $1$, $n-1$, and $m-n$; the determinant then follows from a block-arrowhead calculation in the basis $\{1, H_{1,2}, H_2,\dots,H_n\}$.

What would settle it

Take $n=3$ and let $X$ be the constant random variable $X\equiv 0$. Then for every $d$ all three vectors coincide, all distances are $0$, and $p_d=1$; substituting $\operatorname{Var} X=0$ and $C_1=0$ into formula (3.6) makes the denominator zero, so the asymptotic statement as written fails for this degenerate input.

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Extended reading notes

Core claim

The central claim (Theorem 3.3) is that for iid $d$-dimensional vectors with entries from a lattice distribution $X$, the probability $p_d$ that all $\binom{n}{2}$ pairwise Euclidean distances are equal satisfies, as $d \to \infty$, $$p_d \sim \frac{1}{$d^{{(m-1)/2}}$} \left( m(2\pi)^{m-1}(\operatorname{Var} X)^{2(m-n)}(4(\operatorname{Var} X)^2 + (n-2)C_1)^{n-1} \right)^{-1/2},$$ with $m = \binom{n}{2}$ and $C_1 = \operatorname{Var}((X - \mathbb{E}[X])^2)$. The proof reduces the event to a partial-sum probability for iid increment vectors, applies a multidimensional local limit theorem, and computes the two ingredients explicitly: the lattice spanned by the increments has fundamental volume $2^{m-n}$ for every non-degenerate integer-valued $X$ (after rescaling and taking gcd $1$), and the determinant of the covariance matrix of the increments is evaluated through the spectrum of the adjacency matrix of the line graph of the complete graph $K_n$. The general finite-support case (Theorem 3.8) keeps the same structure after embedding the support into $\mathbb{Q}^{\ell}$.

Load-bearing premise

The asymptotic formula is derived from a multidimensional local limit theorem that needs the increment vectors to have a non-singular covariance matrix and a support spanning a full-rank lattice; the paper verifies this for non-degenerate $X$, but it never explicitly excludes a constant random variable, for which the formula is undefined although the equidistance probability is exactly $1$.

Editorial extensions

If this is right

  • In the binary case $X\sim\operatorname{Ber}(1/2)$, equidistant vectors are exactly equidistant binary codes; the theorem yields their asymptotic density as $\sqrt{2^{3m-2n-1}/(m\pi^{m-1})}\,d^{-(m-1)/2}$.
  • Because the lattice volume factor $2^{m-n}$ is independent of $X$, the main-order constant for any integer-valued $X$ with $\gcd(\operatorname{Supp}X)=1$ depends on $X$ only through $\operatorname{Var}X$ and $C_1$, i.e. its first four moments.
  • For fixed $n$, the probability decays as $d^{-(\binom{n}{2}-1)/2}$: triples become linearly rare in $d$, while larger equidistant families vanish faster.
  • The method extends to $L^p$ distances for integer-valued $X$ and integer $p$, and Theorem 3.8 covers arbitrary finite supports by embedding the support into $\mathbb{Q}^{\ell}$; so the same power law governs a wide family of distance notions.

Reading between the lines

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

  • A natural extension would be to replace 'all distances equal' with a prescribed multiset of distances, e.g. exactly two distinct distance values; the same lattice and covariance machinery would apply to principal submatrices of the same line-graph matrix, and the exponent should again be $d^{-\ell(m-1)/2}$ with the same $\ell$.
  • The explicit eigenvectors of $L(K_n)$ used here also suggest a route to second-order corrections via Edgeworth expansions of multidimensional lattice distributions, refining the '$\sim$' relation to an asymptotic expansion in powers of $d^{-1/2}$.
  • The degenerate-case breakdown indicates that a separate analysis of near-constant distributions (with variance tending to $0$ slowly in $d$) could reveal a crossover regime where the equidistance probability transitions from $1$ to the power law; this regime is not treated in the paper.
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Editorial analysis

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

Referee Report

2 major / 4 minor

Summary. This paper studies the probability p_d that n independent d-dimensional vectors with iid entries from a finite-lattice or finite-support distribution X form an equidistant set under Euclidean distance. The authors rewrite the equidistance condition as a sum of iid difference vectors equal to zero, apply the multidimensional local limit theorem of Krafft, and reduce the problem to two lattice/linear-algebra tasks: computing the fundamental volume of the lattice generated by the support of the increments, and computing the determinant of their covariance matrix. For the binary case they obtain the explicit constant in (3.1); for general lattice X they state the closed-form constant (3.6) in terms of VarX and C1=Var((X-EX)^2); and for arbitrary finite support they state a more general formula (3.18) involving lattice volumes of the sets of products of support elements, using a Hamel-basis embedding.

Significance. If the stated formulas hold under the right hypotheses, the result is a clean and useful high-dimensional asymptotic: it identifies the 1/d^{(m-1)/2} rate and gives explicit constants without free parameters. The proof strategy is original and transparent: the lattice basis for the overlapping map H in Theorem 2.1 is explicit and checkable, the derivation of |Lat V|=2^{m-n} in Lemma 3.2 is constructive, and the determinant computation using the spectral decomposition of the line graph of K_n (Propositions A.1, A.3, A.4) is a nice piece of linear algebra. The general finite-support theorem 3.8 is ambitious and, after a careful check, the covariance identity C0-2C1=4(VarY)^∘2 is valid in the stated vector setting. The main weakness is that Theorem 3.3, as stated, is not invariant under scaling of X and therefore is false without an explicit normalization assumption; this is a localized but load-bearing error in the central theorem.

major comments (2)
  1. [Theorem 3.3, Eq. (3.6)] The statement as written is not invariant under scaling of X, although the event is. In the proof in Section 3.2 the authors first translate so that 0 is in the support, multiply by the lcm to make the support integer, and divide by the gcd so that gcd Supp X = 1; only after that normalization is |Lat V| = 2^{m-n} used. Formula (3.6), however, is expressed through the moments of the original X. Under X -> cX, VarX -> c^2 VarX and C1 -> c^4 C1, so the right-hand side of (3.6) changes by c^{-2(m-1)}, while p_d is unchanged. Concretely, for n=3 and X uniform on {0,2}, the equidistance probability is the same as for Ber(1/2), for which (3.1) gives constant 2/(pi sqrt(3)); substituting VarX=1, C1=0 into (3.6) gives 1/(8 pi sqrt(3)), which is smaller by a factor of 16. The theorem must either state the formula for the normalized variable (integer support with gcd 1 after translation), or include the missing lattice-volume/normalization factor explicitly.
  2. [Theorem 3.3 and Theorem 3.8, hypotheses] The theorem does not exclude a degenerate lattice distribution: if X is a constant, then p_d=1 for every d, but VarX=0 and C1=0, so the right-hand side of (3.6) is undefined (division by zero), and the covariance matrix in the local CLT is singular. Add an assumption such as |Supp X| >= 2 or VarX > 0 to Theorem 3.3; Theorem 3.8 similarly needs a non-degeneracy assumption ensuring positive-definite covariance and a positive dimension ell.
minor comments (4)
  1. [Proof of Lemma 3.2] The first line says 'in particular Theorem 3.3' where the intended reference is Theorem 2.1, since the basis of Lat H is being used.
  2. [Proof of Theorem 3.8] After (3.19), the displayed definitions 'C0 := (Var(Y1-Y2))^2' and 'C1 := Cov(Y1-Y2, Y1-Y3)' do not match the theorem statement; they should read C0 := Var((Y1-Y2)^2) and C1 := Cov((Y1-Y2)^2, (Y1-Y3)^2). As written, the types of these objects are unclear.
  3. [Proof of Theorem 3.8] The word 'denumerator' should be 'denominator' in the sentence after (3.19).
  4. [Abstract and Introduction] The abstract claims the result for all finitely supported X with an explicit constant in terms of the first four moments; for the finite-support case the statement in Theorem 3.8 also depends on lattice volumes |Lat X1| and |Lat X2|, so the abstract should be phrased to avoid suggesting that first moments alone determine the constant in the general finite-support setting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic constant is a closed-form function of moments plus a derived lattice volume, with the only external input the standard multidimensional local limit theorem.

full rationale

The derivation chain is self-contained and non-circular. The probability p_d is first rewritten as P(\sum_{l=1}^d V_l = 0) (eqs. 3.2, 3.7, 3.8), which is not an identity with the target asymptotic but an exact reformulation of the event. The asymptotics then come from the multidimensional local limit theorem of Krafft (eq. 3.3/3.7), an external result whose hypotheses (non-singular covariance, full-rank lattice) are verified for non-degenerate X. The lattice volume |Lat V| = 2^{m-n} is computed, not assumed, from an explicit basis of the overlapping map H (Theorem 2.1, Lemma 3.2) and a Bézout/gcd argument (eq. 3.9). The covariance determinant is computed by linear algebra using the eigenstructure of the line graph of K_n, with the relevant eigenvalues and eigenvectors derived in Proposition A.1 and the block determinant computed in Propositions A.3–A.4. No fitted parameter is introduced, no quantity is defined in terms of the target probability, and no load-bearing conclusion is justified by a citation to the authors' own prior work. The only identified issues—the omitted normalization/scaling hypothesis in the statement of Theorem 3.3 for arbitrary finite lattice X and the degenerate Var X = 0 case—are correctness or hypothesis gaps, not circularity, because they do not make the claimed result equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No parameters are fitted; the constant is a function of moments. The derivation relies on the local CLT and on a chosen Q-basis for non-lattice finite supports. The main implicit assumption is non-degeneracy of X.

assumptions (3)
  • standard math Multidimensional local limit theorem for lattice distributions (Krafft 1967), used as eq. (3.3).
    The central asymptotic (3.7) and (3.19) depend on this theorem; it is cited but not proved.
  • domain assumption The random variable X has finite lattice support and is non-degenerate (Var X > 0).
    Not stated explicitly in the theorems; formula (3.6) requires Var X > 0 to be finite and non-zero.
  • domain assumption For the general finite-support case, the Q-span of the support is embedded into Q^ell via a chosen Hamel basis phi.
    Theorem 3.8 fixes a reference basis Bref; the asymptotics are expressed in terms of determinants and lattice volumes relative to this basis, and the proof assumes the local CLT applies in the embedded coordinates.

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Pith. "Pith review of On the probability of n equidistant points in high-dimensional lattices." pith.science (2026). https://pith.science/paper/P33LUMYD

@misc{pith2026250203440,
  author       = {Pith},
  title        = {Pith review of: On the probability of n equidistant points in high-dimensional lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P33LUMYD}},
  note         = {Machine review of arXiv:2502.03440}
}
abstract

Consider $n$ $d$-dimensional vectors with iid entries from a lattice distribution $X$. We show that the probability that all distances between them are equal is asymptotically \[ C_n\cdot\frac{1}{d^{(m-1)/2}} \quad \text{for} \quad d \to \infty \quad \text{and} \quad m = \binom{n}{2}, \] with an explicit constant in terms of the first 4 moments of $X$. Moreover, we generalise this result to encompass all finitely supported $X$, as well as under different distances. Our method relies on the relatively rarely used multidimensional local limit theorem and an analysis of the lattice on $\mathbb{Z}^{\binom{n}{2}}$ spanned by the image of the \emph{overlapping} map \[ H : \{0,1\}^n \to \{0,1\}^{\binom{n}{2}}, \quad (v_1, \dots, v_n) \mapsto \Bigl( \mathbf{1}_{\{v_i \neq v_j\}} \Bigr)_{1 \le i < j \le n}. \]

Figures

Figures reproduced from arXiv: 2502.03440 by the authors.

Figure 1
Figure 1. The vectors H([1 : 1]), H([1 : 2]), H([1 : n − 2]) and H([1 : n − 1]) with empty cells representing 0s. We recall that we put a one in H(I) for every (i, j) such that i ∈ I, j /∈ I or vice versa. Theorem 2.1. Let H be the map from (1.3) or alternatively defined for a set I ⊂ [1 : n] by H(I) = X i∈I,j /∈I ei,j . Then the vectors • H([1 : 1]), H([1 : 2]), . . . , H([1 : n − 1]) • alongside all of the form 2ei,j with j… view at source ↗
Figure 2
Figure 2. Illustrating the relation H({i}) + H({j}) − H({i, j}) = 2ei,j . On the left: H({i}) is shown in green and H({j}) in red; on the right: H({i, j}). and H([i + 1 : j] = H([1 : j]) − H([1 : i]) + 2X i k=1 X j ℓ=i+1 ek,ℓ. (2.3) In particular, equation (2.3) shows how to express the image of an interval in terms of H([1 : 1]), . . . , H[1 : n − 1], H[1 : n] = 0 and these of type 2ei,j . We note that we have picked in the … view at source ↗
Figure 3
Figure 3. Illustrating the relation H([i + 1 : j]) = H([1 : j]) − H([1 : i]) + Pi k=1 Pj ℓ=i+1 2ek,ℓ. On the left: H([1 : i]) is shown in green and H([1 : j]) in red; on the right: H([i + 1 : j]). Therefore, applying (2.1), H(I) = H(∪Ii) = P i H(Ii) − P k,ℓ 2ek,ℓ, which shows that the vectors from the theorem statement generate Lat ImH. To see they are linearly independent, we once again refer to [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The vectors H∗ e ([1 : 1]), H∗ e ([1 : 2]), . . . H∗ e ([1 : n − 2]) and H∗ e ([1 : n − 1]), with the embedded cell shaded in grey. Only the first vector in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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