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arxiv: 2607.05374 · v1 · pith:YUHTKQZP · submitted 2026-07-06 · math.CO · math.MG· math.NT

The Minkowski grid has robustly many repeated distances

Sungchul Lee , Cosmin Pohoata , Daniel G. Zhu This is my paper

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classification math.CO math.MGmath.NT
keywords deltagtrsimpropertysizesubsetsconjecturecontaindistance
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The pith

Minkowski grid forces repeated distances in every large subset

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

The paper proves that for every n, one can construct an n-point set P in the plane (a Minkowski grid derived from a tower of totally real number fields of bounded root discriminant) such that every subset A of P has a single distance occurring at least |A|^2 / n^{1-delta} times among its ordered pairs, for some absolute constant delta > 0. This is a 'robust' property: it holds not just for the full set P but for every subset, no matter how small. The key mechanism is vertical amplification -- instead of working over the integers (the ordinary square grid), the construction works over the ring of integers of a high-degree number field where many primes congruent to 1 mod 4 split completely. Each splitting prime provides a factor of 2 in the distance-multiplicity gain, and because the field degree grows exponentially while the root discriminant stays bounded, these gains compound to a polynomial improvement over the trivial pigeonhole bound. The proof bypasses class group considerations used in prior constructions: it relies on a combinatorial large sieve (factoring x^2 + y^2 modulo split primes via the Chinese remainder theorem) combined with lattice counting estimates for symmetric Minkowski boxes.

Core claim

The central object is the Minkowski grid -- the image of a symmetric box {alpha in O_K : |sigma_j(alpha)| <= X for all embeddings sigma_j} in the ring of integers of a totally real number field K, embedded into R^2. The key identity is that for primes p = 1 mod 4 splitting completely in K, the congruence x^2 + y^2 = 0 mod p factors as (x + iota*y)(x - iota*y) = 0 mod p, where iota is a square root of -1. Over a degree-d field, each such prime splits into d prime ideals, yielding 2^d sign choices per prime. With k such primes, the sieve produces a factor of 2^{kd} in the distance-multiplicity lower bound, while the bounded root discriminant ensures that lattice-counting losses remain bounded,

What carries the argument

Combinatorial large sieve over number field rings; Minkowski lattice embedding and box counting; bounded-discriminant towers of totally real number fields with completely split 1 mod 4 primes; Chinese remainder theorem for prime ideal factorizations of x^2 + y^2

If this is right

  • Every n-point planar set of this type has a distance occurring at least n^{1+delta} times, so a rescaled copy is a counterexample to the Erdos unit distance conjecture with a simpler proof that avoids class groups.
  • Every subset of size at least n^{1-delta} must contain an isosceles triangle, confirming Erdos's 1980 conjecture that there exist n-point sets where no subset larger than n^{1-delta} avoids isosceles triangles.
  • Every subset of size at least n^{1/2-delta} must contain a repeated distance, answering negatively the question of whether f_dist(n) = n^{1/2 - o(1)}.
  • The construction simultaneously achieves all three properties (unit distance multiplicity, isosceles triangle forcing, repeated distance forcing) in a single point set, which was not previously known.
  • The robust property (holding for every subset, not just the full set) distinguishes this from prior unit-distance counterexamples and connects the construction to extremal hypergraph independence number problems.

Load-bearing premise

The proof depends on the existence of an infinite tower of totally real number fields with bounded root discriminant and infinitely many primes congruent to 1 mod 4 that split completely in every field of the tower. This is argued as a modification of a known result, with the additional 1 mod 4 splitting condition justified in a few sentences via a density theorem. If this modification does not hold as stated -- for instance, if achieving more split primes requires a larger (

What would settle it

If the bounded root discriminant D in the tower of totally real number fields cannot be made independent of the number of completely split 1 mod 4 primes required, then the parameter selection (choosing k primes so that the multiplicative gain 2^k exceeds 2D) fails, and the polynomial improvement collapses to the sub-exponential Ramanujan-type bound of Theorem 1.

read the original abstract

We show that there exists a constant $\delta > 0$ such that for any positive integer $n$ there exists a set of $n$ points $P \subset \mathbb{R}^2$ with the following property: for every subset $A \subseteq P$ of size $|A| \geq 2$, \[ \max_{\lambda>0} \#\{(a,b)\in A \times A: a\ne b,\ \lvert a-b\rvert=\lambda\} \gtrsim \frac{|A|^2}{n^{1-\delta}}.\] Our result is a vertical amplification of a robust Ramanujan estimate recently established by Croot-Mao-Pohoata-Sheffer-Yip for arbitrary subsets of the ordinary square grid, and is inspired by recent constructions for the Erd\H{o}s unit distance problem and the Elekes-R\'onyai problem. Taking $A=P$, the inequality above gives a distance occurring $n^{1+\delta}$ times in $P$; thereby a scaled copy of $P$ is a counterexample for the unit-distance conjecture. In addition, the same inequality shows that (1) all subsets of $P$ of size $\gtrsim n^{1-\delta}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n^{1/2-\delta}$ must contain repeated distances. These features give polynomially improved estimates for old problems of Erd\H{o}s. The existence of a set satisfying property (1) confirms a conjecture of Erd\H{o}s from 1980, whereas the existence of a set with property (2) answers a question of Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky in the negative.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 7 minor

Summary. The paper proves that there exists δ>0 such that for every n there is an n-point set P⊂ℝ² with μ(A)≳|A|²/n^{1−δ} for every subset A⊆P of size ≥2, where μ(A) is the maximum ordered-pair multiplicity of a single nonzero distance. The construction uses a Minkowski grid from a totally real number field of growing degree with bounded root discriminant, building on the combinatorial large sieve of Croot–Mao–Pohoata–Sheffer–Yip and the vertical amplification framework. Consequences include a polynomial improvement for the unit distance problem, confirmation of Erdős's 1980 conjecture on isosceles triangles in large subsets, and a negative answer to a question of Conlon–Fox–Gasarch–Harris–Ulrich–Zbarsky about f_dist(n).

Significance. The results represent a genuine advance on several classical problems simultaneously. The proof of Lemma 6 is the core technical contribution: the coset counting via CRT, the Cauchy–Schwarz lower bound on |S_ε|, the divisibility argument for q(a,b), and the lattice counting via Lemma 5 are all traceable and clean. The parameter selection in the proof of Theorem 2 is explicit. The approach bypasses class group considerations used in prior constructions, which is a notable simplification. The paper provides falsifiable, quantitative predictions and the construction is concrete modulo the algebraic number-theoretic input.

major comments (3)
  1. Proposition 4 (§2): The proof claims a modification of [HMR21, Theorem 4] yields an infinite tower of totally real 2-extensions with bounded root discriminant D and infinitely many 1 mod 4 primes splitting completely in every field of the tower. The 1 mod 4 condition is imposed in two sentences via Chebotarev applied to Q̄^H(i). The concern raised in the stress-test note—that the argument shows density of 1 mod 4 primes in each quotient Gal(K_i/Q) but does not obviously produce a single infinite set splitting completely in the full extension L/Q—appears to be resolved by the observation that L is totally real and hence linearly disjoint from Q(i), so primes splitting completely in L automatically split in Q(i) and are thus 1 mod 4. However, the manuscript does not state this linear disjointness argument explicitly. The proof as written says 'every open normal subgroup H of Gal(Q̄/Q)' but
  2. Proposition 4 (§2): The manuscript states 'Note that unlike [Alo+26], we need the root discriminant to be bounded independently of the number of completely split primes.' This is the load-bearing requirement: the proof of Theorem 2 fixes k primes at the start and then lets d→∞, so D must not depend on k. The proof of Proposition 4 does not explicitly verify this independence. The application of [HMR21, Theorem 4] with S={3,5,7,11,13,17,∞} produces a tower with bounded root discriminant, but the set of completely split primes is whatever the tower gives—it is not clear from the argument as written that one can prescribe an arbitrary finite subset of 1 mod 4 primes to split completely while keeping D bounded independently of the size of that subset. The manuscript should clarify whether the tower from [HMR21, Theorem 4] already has infinitely many completely split primes (in which case one
  3. Proof of Theorem 2 (§3): The condition |A|≥2Q^d is required for Lemma 6, and the final argument handles |A|<n^{1/2−δ₁} by the trivial bound μ(A)≥1. The constants δ₁ and δ₂ are defined by (100Q²)^{1/2−δ₁}=2Q and (10000Q⁴)^{δ₂}=2. Since Q depends on the choice of k (which is fixed at the start), δ=min(2δ₁,δ₂) is a fixed positive constant. This part of the argument is correct, but the manuscript should state more explicitly that δ is independent of n (it depends only on k, D, and c, all of which are fixed before n enters).
minor comments (7)
  1. Title: 'ROBUSTL Y' and 'REPEA TED DIST ANCES' have spacing artifacts from the PDF extraction; presumably fine in the actual manuscript but worth checking.
  2. §1, proof sketch: 'The proof of Theorem 1 relies on...' — this should probably say 'The proof of Theorem 1 in [CMPSY26] relies on...' for clarity, since Theorem 1 is not proved in this paper.
  3. §2, Proposition 4 proof: 'selecting distinct primes whose Frobenius elements fall in an infinite sequence of open normal subgroups' — the referent of 'selecting P' is ambiguous (P is later used for the set of primes); consider renaming.
  4. §3, proof of Theorem 2: 'By setting constants appropriately it suffices to consider n≥100Q²' — specify which constants and why this reduction is valid.
  5. §3, proof of Theorem 2: The choice X=n^{1/(2d)}√D should be cross-checked against the condition X≥10Q used in Lemma 6; the manuscript asserts this but the verification is implicit.
  6. Lemma 6: The constant c=1/34 is derived at the end; the statement says 'for some absolute constant c>0'. Consider stating c=1/34 in the lemma for concreteness.
  7. The reference [OptCon] and [ErdPro] use non-standard citation formats (URLs without arXiv identifiers); ensure consistency with journal style.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful reading and for identifying two genuine gaps in the exposition of Proposition 4, along with one clarification request regarding the proof of Theorem 2. All three comments are well-taken and will be addressed in the revision.

read point-by-point responses

  1. Referee: Proposition 4 (§2): The proof does not explicitly state the linear disjointness argument (L totally real, hence linearly disjoint from Q(i), so primes splitting completely in L are automatically 1 mod 4).

    Authors: The referee is correct that the linear disjointness argument is implicit but not stated. Since L is totally real and Q(i) is imaginary quadratic, L and Q(i) are linearly disjoint over Q. Therefore any prime splitting completely in L also splits completely in Q(i), hence is 1 mod 4. We will add this argument explicitly to the proof of Proposition 4. revision: yes

  2. Referee: Proposition 4 (§2): The manuscript does not verify that the root discriminant bound D is independent of the number of completely split primes. The proof of Theorem 2 fixes k primes and then lets d→∞, so D must not depend on k.

    Authors: The referee has identified a genuine gap in the exposition. The argument as written does not make clear that D is independent of k. The resolution is as follows. The tower L/Q from [HMR21, Theorem 4] is constructed once and for all, yielding a fixed constant D bounding all root discriminants. This tower already has infinitely many completely split 1 mod 4 primes (by the linear disjointness argument above). In the proof of Theorem 2, we select k primes from this pre-existing infinite set P; this selection does not modify the tower or affect D. Thus D is fixed before k is chosen, and k is fixed before n (hence d) enters. We will revise the proof of Proposition 4 to make clear that the tower and the constant D are constructed independently of the set P of completely split primes, and that P is infinite. We will also add a sentence in the proof of Theorem 2 clarifying the logical order: the tower (and hence D) is fixed first, then k primes are chosen from P, then d grows with n. revision: yes

  3. Referee: Proof of Theorem 2 (§3): The manuscript should state more explicitly that δ is independent of n (it depends only on k, D, and c, all fixed before n enters).

    Authors: The referee's observation is correct, and we agree that the independence of δ from n should be stated more explicitly. The constants δ₁ and δ₂ are defined by (100Q²)^{1/2−δ₁} = 2Q and (10000Q⁴)^{δ₂} = 2, where Q = p₁p₂···p_k depends only on the fixed k. Since k, D, and c are all fixed before n enters the argument, δ = min(2δ₁, δ₂) is indeed a fixed positive constant independent of n. We will add an explicit sentence to this effect in the proof of Theorem 2. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-citations are used as black-box inputs with stated assumptions, not as the target result.

full rationale

The paper's central derivation (Lemma 6 and the proof of Theorem 2) is self-contained. Lemma 6 takes as inputs: (a) the number-field tower from Proposition 4, which is attributed to Hajir-Maire-Ramakrishna [HMR21, Theorem 4] with a stated modification, and (b) the lattice estimates from Lemma 5, whose first inequality is attributed to [BSSZ26, Lemma 3.3] and [Poh26, Lemma 3.2] but whose second inequality is proven in-line. The proof of Theorem 2 then combines Lemma 6 with parameter selection (choosing k primes, setting d, X, etc.) in a way that is not tautological: the bound μ(A) ≥ |A|² · (2D/X²)^d = |A|² · 2^d/n follows from the interplay between the lattice counting, the 2^kd factor from the sign-vector decomposition, and the divisibility argument, none of which is a renaming of the conclusion. The self-citations to [CMPSY26] (Theorem 1, co-authored by Pohoata) and [Poh26] (sole-authored by Pohoata) are used as starting points or for the amplification heuristic, not as the target result being proven. The target result (Theorem 2) is strictly stronger than Theorem 1 and is derived via the new Lemma 6. The concern about Proposition 4's Chebotarev argument is a correctness risk (whether the modification of [HMR21] is valid), not a circularity issue: the proposition is stated with explicit assumptions and attributed to an external theorem, and if those assumptions hold, the downstream derivation is independent. No step in the derivation chain reduces to its inputs by definition or by a fitted parameter renamed as a prediction. The minor self-citations are standard and do not undermine the independence of the central argument. Score: 2 (one minor self-citation chain that is not load-bearing for the central claim).

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The paper introduces no new mathematical objects. The Minkowski grid construction uses standard objects from algebraic number theory (rings of integers, Minkowski embeddings, ideal lattices). The tower of number fields is from [HMR21]. No new particles, forces, dimensions, or postulated entities are introduced.

free parameters (3)
  • δ = not explicitly computed; exists by construction
    δ = min(2δ₁, δ₂) where δ₁, δ₂ are defined by (100Q²)^{1/2−δ₁} = 2Q and (10000Q⁴)^{δ₂} = 2. Q depends on k, which is chosen so the product of 2p_i/(p_i+1) terms ≥ 2D. The value is not numerically specified.
  • D (root discriminant bound) = not explicitly computed; exists by [HMR21]
    D is the bound on root discriminants rd(K_i) from the tower in Proposition 4, attributed to Hajir-Maire-Ramakrishna. Treated as an input constant.
  • k (number of split primes) = chosen so c · ∏(2p_i/(p_i+1))^d ≥ 2D
    k is fixed once at the start of the proof and does not grow with n. The condition is satisfiable because 2p/(p+1) ≥ 5/3 and P is infinite.
axioms (4)
  • standard math Hajir-Maire-Ramakrishna theorem: there exists an infinite totally real Galois pro-2 extension L/Q of bounded root discriminant with infinitely many rational primes splitting completely.
    Proposition 4, Section 2. Cited as [HMR21, Theorem 4]. Published result in algebraic number theory.
  • domain assumption The modification of [HMR21] to additionally impose the 1 mod 4 condition on split primes via Chebotarev density applied to Q̄^H(i).
    Proposition 4, Section 2. Argued in two sentences. The underlying Chebotarev theorem is standard; the application to the specific pro-2 extension is plausible but not fully verified here.
  • domain assumption The combinatorial large sieve estimate (Theorem 1 / [CMPSY26]) for subsets of the ordinary square grid.
    Section 1, Theorem 1. Used as motivation and conceptual foundation; the present paper's Theorem 2 is a vertical amplification of this result.
  • standard math Lattice counting estimates for Minkowski boxes (Lemma 5).
    Section 2, Lemma 5. The lower bound is attributed to [BSSZ26, Lemma 3.3] and [Poh26, Lemma 3.2]; the upper bound is proven in-line.

pith-pipeline@v1.1.0-glm · 12265 in / 3519 out tokens · 201724 ms · 2026-07-07T14:02:29.860735+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 7 internal anchors

  1. [1]

    Remarks on the disproof of the unit distance conjecture

    arXiv:2605.20695. [Beh46] F. A. Behrend. On sets of integers which contain no three terms in arithmetical progression. Proc. Nat. Acad. Sci. U.S.A.32, 331–332

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Balogh and J

    [BS18] J. Balogh and J. Solymosi. On the number of points in general position in the plane.Discrete Anal.2018:16

    work page 2018

  3. [3]

    The sum-product conjecture is false for real numbers

    arXiv:2605.28781. [CEWW24] F. Charton, J. S. Ellenberg, A. Z. Wagner, and G. Williamson. PatternBoost: Constructions in Mathematics with a Little Help from AI. Preprint

    work page internal anchor Pith review Pith/arXiv arXiv

  4. [4]

    PatternBoost: Constructions in Mathematics with a Little Help from AI

    arXiv:2411.00566. [CFGHUZ15] D. Conlon, J. Fox, W. Gasarch, D. G. Harris, D. Ulrich, and S. Zbarsky. Distinct volume subsets.SIAM J. Discrete Math.29, 472–480

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    A combinatorial large sieve for Sidon sets, distances, and norm forms

    arXiv:2606.17487. [EG70] P. Erdős and R. K. Guy. Distinct distances between lattice points.Elem. Math.25, 121–123

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    [ErdPro] T. F. Bloom. Erdős Problems. 2026.url:https://www.erdosproblems.com. [ET41] P. Erdős and P. Turán. On a problem of Sidon in additive number theory, and on some related problems.J. London Math. Soc.16, 212–215

    work page 2026

  7. [7]

    [Lee26] S. Lee. Planar point sets with small distance-distinct subsets. Preprint. 2026.url:https://g ithub.com/lsngchl/Erdos1208. [LT95] H. Lefmann and T. Thiele. Point sets with distinct distances.Combinatorica15, 379–408

    work page 2026

  8. [8]

    [Mil20] J. S. Milne. Algebraic Number Theory. 2020.url:https://www.jmilne.org/math/CourseNo tes/ANT.pdf. [Ope26] OpenAI. An OpenAI model has disproved a central conjecture in discrete geometry

    work page 2020

  9. [9]

    [OptCon] D

    url:https://openai.com/index/model-disproves-discrete-geometry-conjecture/. [OptCon] D. Davis, P. Ivanisvili, T. Tao, and contributors. Optimization Constants in Mathematics. 2026.url:https://teorth.github.io/optimizationproblems. [Poh26] C. Pohoata. Split primes and the Elekes-Rónyai problem. Preprint

    work page 2026

  10. [10]

    Split primes and the Elekes-R\'onyai problem

    arXiv:2606.13619. [Ram15] S. Ramanujan. Highly composite numbers.Proc. London Math. Soc.2nd ser.14, 347–409

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    An explicit lower bound for the unit distance problem

    arXiv: 2605.20579. [She14] A. Sheffer. Distinct Distances: Open Problems and Current Bounds. Preprint

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    Distinct Distances: Open Problems and Current Bounds

    arXiv: 1406.1949. [Sin38] J. Singer. A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc.43, 377–385

    work page internal anchor Pith review Pith/arXiv arXiv 1949

  13. [13]

    [Tao26] T. Tao. A digestion of unit distance constructions. 2026.url:https://terrytao.wordpress .com/2026/07/03/a-digestion-of-unit-distance-constructions/

    work page 2026