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On subsets of lattice cubes avoiding affine and spherical degeneracies

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Pith's one-line read For large n, an n×n grid contains at least 7n/12 points with no four on a common circle or line.

desk verdict New 7n/12 bound for no-four-on-a-circle is a real step, but the proof's key transfer from Huxley–Konyagin is asserted, not demonstrated. read the letter →

arxiv 2509.06935 v1 pith:RWDPBMSS submitted 2025-09-08 math.CO

Anubhab Ghosal , Ritesh Goenka , Peter Keevash This is my paper
classification math.CO MSC 05D4052C1052C35
keywords no-four-on-a-circlelatticecubesaffinedegeneraciessphericaldeletionmethodcyclicquadrilateralsintegralmatricesoffixedrankpoint-flatincidences
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 studies how many points one can choose from the integer cube [n]^d while avoiding geometric coincidences: r points in a k-dimensional affine or linear subspace, or d+2 points on a (d−1)-dimensional sphere. Its central new result is an asymptotic count of cyclic quadrilaterals in the square grid [n]^2: there are γ n^5, with γ between 0.35974 and 0.36017, up to an error term of order n^{4+18/29+ε}. Feeding this count into the deletion method gives a randomized construction of at least 7n/12 grid points with no four collinear or concyclic, improving the previous n/4 bound. The same counting-plus-deletion strategy gives new lower bounds for avoiding affine and linear degeneracies when 1

What carries the argument

The engine is the deletion method for r-uniform hypergraphs: if a hypergraph has v vertices and e edges, removing vertices from a random subhypergraph of minimum degree yields an independent set of size about v^{r/(r−1)}/e^{1/(r−1)}. Each extremal problem is therefore reduced to counting edges in the right hypergraph. For affine and linear degeneracies, the count is of d×r integer matrices of rank at most k, controlled by an asymptotic theorem on integral matrices of fixed rank. For cyclic quadrilaterals, the count splits into isosceles trapezia—counted exactly via a lattice-point lemma for convex polygons, producing the constant γ—and asymmetric quadrilaterals, bounded by a transferred esti

What would settle it

Enumerate all asymmetric cyclic quadrilaterals in [n]^2 (quadruples that are not isosceles trapezia) for n as large as feasible and fit their growth; if their count is Θ(n^5) rather than O(n^{4+18/29+ε}), Lemma 4.2 is false and Theorem 1.3's claimed error term collapses. Independently, evaluating the constant γ by truncating the sum in (12) should place it in (0.35974, 0.36017); any reliable fit outside this interval would refute Corollary 1.4's constant.

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Extended reading notes

Core claim

On its own terms, the paper establishes that almost all cyclic quadrilaterals in the square lattice are symmetric, namely isosceles trapezia. It counts these explicitly, obtaining γ n^5 + O(n^4 log n) with the constant γ given by a convergent number-theoretic sum, and it combines this with a transferred number-theoretic estimate to show that the remaining asymmetric quadrilaterals are only O(n^{4+18/29+ε}). Summing gives Theorem 1.3: the number of cyclic quadrilaterals in [n]^2 is γ n^5 + O(n^{4+18/29+ε}). After subtracting collinear quadruples, the deletion method yields Corollary 1.4: for large n, one can choose at least 7n/12 points with no four collinear or concyclic. For the higher-dime

Load-bearing premise

The numerical conclusion 7n/12 rests on an unproved transfer: a known bound on asymmetric cyclic quadrilaterals under a bounded-circumradius condition is asserted, by 'a closer inspection', to hold under a bounded-diameter condition, and if the true asymmetric count in [n]^2 were larger than O(n^{4+18/29+ε}), the error term in Theorem 1.3 could exceed the margin needed for the constant 7/12.

Editorial extensions

If this is right

  • No-four-on-a-circle: for large n, f_circ(n) ≥ 7n/12, a constant-factor improvement over the previous n/4 lower bound, obtained by a probabilistic construction rather than an algebraic one.
  • No cospherical points: for every d≥3, f_sph(n,d) = Ω(n^{min{d,4}/(d+1) − c/log log n}), improving the previous lower bound for d≥4; for d=3,4 this is within a subpolynomial factor of the conjectured n^{d/(d+1)}.
  • Affine degeneracies: for 1<k<d−1 and r>d+1, f_aff(n,d,k,r) = Θ(n^{d−k}), matching the trivial upper bound and extending the previously known range r>dk.
  • Linear degeneracies: for k<d and r≥k+1, f_lin(n,d,k,r) is determined up to polylog factors in new regimes, including the case k=d−1 where it recovers the known n^{d/(d−1)} order.
  • The asymptotic count of cyclic quadrilaterals, γ n^5 with γ≈0.36, settles the order of that quantity and provides a benchmark for any future construction or upper bound.

Reading between the lines

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

  • If the transferred diameter bound for asymmetric cyclic quadrilaterals can be proved in full, the only remaining gap to a fully unconditional Theorem 1.3 is the computer-assisted enclosure of γ; a purely analytic evaluation of the sum defining γ could push the 7n/12 constant higher.
  • The random-matrix analogy in the paper suggests a route to its Conjecture 5.1: if the dominant singularity events for the (d+2)×(d+2) matrix are zero rows or columns and equal rows or columns, then S(n,d)=O(n^{d^2+d}), nearly matching the lower bound n^{d^2+d−2}.
  • The linear-size probabilistic no-four-circle set indicates that extremal configurations need not be algebraic; extending a similar random construction to the no-three-in-line problem would speak to whether large no-three-in-line sets must reduce to an algebraic curve modulo some prime.
  • The same deletion-plus-counting pipeline could be applied to avoiding five or more concyclic points, using the already-derived counts of isosceles trapezia and of collinear r-tuples.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies extremal problems on subsets of [n]^d avoiding affine, linear, and spherical degeneracies. For affine and linear degeneracies, Theorems 1.1 and 1.2 improve earlier bounds of Sudakov–Tomon and Lefmann by applying Spencer's deletion lemma to counts of rank-deficient matrices obtained from Katznelson's theorem. For spherical degeneracies, Theorem 1.5 and Corollary 1.6 bound S(n,d) and improve on Suk–White via Lund's incidence bound and lattice-point estimates on spheres. The central new result is Theorem 1.3, which asserts that the number of cyclic quadrilaterals in [n]^2 is γ n^5 + O(n^{4+18/29+ε}) with an explicit constant γ, leading to Corollary 1.4: f_circ(n) ≥ 7n/12, improving Thiele's n/4 lower bound. The proof splits cyclic quadrilaterals into isosceles trapezia, counted in Lemma 4.4, and asymmetric quadrilaterals, bounded via Huxley–Konyagin in Lemma 4.2.

Significance. If the main claims hold, this is a substantial contribution: it gives the first asymptotic count of cyclic quadrilaterals in a square lattice with an explicit constant, improves a twenty-year-old lower bound by a concrete factor, and extends the power of the deletion method to new counting problems. The paper is commendably parameter-free: the constant γ is defined by an explicit convergent sum, not fitted to any target conclusion. The higher-dimensional spherical bound and the clean use of external incidence results are also valuable. The main caveats are that the Huxley–Konyagin transfer in Lemma 4.2 is asserted rather than proved, and the computer-assisted summation in Lemma 4.4 is not auditable from the manuscript. These are local, fixable gaps, but they are load-bearing for the headline theorem.

major comments (3)
  1. [§4.2, Lemma 4.2] This lemma is the only bound on asymmetric cyclic quadrilaterals and is therefore load-bearing for Theorem 1.3 and Corollary 1.4. The proof is the one-sentence claim that a closer inspection of Huxley–Konyagin's proof yields a diameter version because their proof 'only uses the O(R) bound on the distances between the vertices'. The cited theorem, as described, bounds equivalence classes with circumradius at most R, and a diameter bound does not automatically imply a circumradius bound: four almost-collinear lattice points can have small diameter and arbitrarily large circumradius. If the true asymmetric count has any positive n^5 coefficient, the margin in Corollary 1.4 collapses (c=0.51983 vs the required c<0.53134). Please supply a complete proof of Lemma 4.2 or a precise citation to the exact equations in [26] that establish the diameter version.
  2. [§4.2, Lemma 4.4] The computation of the constant γ hinges on the formula for f(a,b). The manuscript states that after plugging the area formula (13) into the sum, 'the above expression simplifies to f(a,b)·m^5 + ...', with the simplification performed in Mathematica and no derivation shown. A single algebraic error in f(a,b) would change γ and could invalidate the 7n/12 constant in Corollary 1.4. The authors should include the summation in an appendix, or provide the Mathematica code/algebraic steps so the simplification can be verified by a referee or reader.
  3. [§3.3, Theorem 1.5] The definition of S(n,d) in the introduction and in Theorem 1.5 explicitly treats hyperplanes as degenerate spheres. However, the upper-bound proof partitions tuples by their 'spherical span' and appears to handle only ordinary (finite-radius) spheres. The hyperplane contribution is first mentioned after the proof, when deriving Corollary 1.6. As written, the proof of the upper bound in Theorem 1.5 is incomplete unless the hyperplane contribution is separately incorporated into the statement of Theorem 1.5. Please reconcile the definition, the proof, and the later use in Corollary 1.6.
minor comments (4)
  1. [§4.2, Lemma 4.5] The rigorous numerical bound γ ∈ (0.35974,0.36017) is obtained by computing the partial sum s_5500 'to an accuracy of 10^-5' in Mathematica. For full reproducibility, specify the algorithm (interval arithmetic? rigorous error bounds?) and include the code or output.
  2. [Proposition 4.1 / Lemma 4.4 / Corollary 1.4] The paper switches between ordered tuples and unordered quadrilaterals without always saying which is meant. For instance, Proposition 4.1 counts ordered r-tuples, while Corollary 1.4 speaks of unordered collinear quadruples. Please state the convention explicitly in each statement and check that the factors of r! are consistent.
  3. [§3.1, Lemma 3.1] In the d=2 base case, the proof assumes the ellipse contains at least 5 points and then solves a 5×5 linear system. If the ellipse is degenerate (e.g., a line or a pair of lines), the conclusion is still true, but the proof as written does not handle these cases. This is a minor gap that can be fixed by a sentence.
  4. [§1, Theorem 1.1] The notation f_d,k,r(n) and g_d,k,r(n) is reused in the 'In particular' clauses with different meanings (f_d,k(n) and g_d,k(n)); this should be disambiguated, as the subscripts differ but the reader may be confused by the same letters.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the counting constants and lower bounds are derived from explicit geometric sums plus external theorems (Katznelson, Lund, Huxley–Konyagin, Spencer); the one asserted step (diameter transfer in Lemma 4.2) is a correctness risk, not a circular reduction.

full rationale

The paper's derivation chain is self-contained in the sense required by the circularity pass: no equation reduces to its own input, no parameter is fitted to a target result, and there are no load-bearing self-citations. (1) Theorem 1.3's constant γ is defined by the explicit coprimality sum (12) in Lemma 4.4, obtained by an area computation for isosceles trapezia with a given axis of symmetry. The constant is computed, not fitted; Lemma 4.5 gives a rigorous interval (0.35974, 0.36017) from a partial sum s_5500 = 0.09309 with accuracy ε = 10^-5 and an explicit tail bound 2.2/N. Corollary 1.4 then uses the bound γ + 7π²/(360ζ(3)) ≤ 0.51983 < 0.53134; the inequality is a genuine arithmetic fact, not an artefact of tuning γ to make 7/12 hold. (2) The affine and linear counting results (Propositions 1.8, 1.9) are reduced to Katznelson's external rank-counting theorem via Theorem 2.1; the paper notes the translation step and gives matching lower bounds. No fitted parameters are involved. (3) The sphere-counting upper bound (Theorem 1.5) uses Lund's external incidence bound (Lemma 3.3) together with internally proved lattice-point bounds (Lemmas 3.1, 3.2) built on the divisor bound; the lower bound is a direct construction. Again, no step reduces to its conclusion. (4) The only notable asserted step is Lemma 4.2, which transfers Huxley–Konyagin's bound from circumradius-bounded equivalence classes to diameter-bounded sets via 'a closer inspection of Huxley and Konyagin's proof' (Section 4.2). This is an omitted proof of a strengthening of an external theorem and is a genuine correctness risk: if the asymmetric-quadrilateral count were c n^5, the margin in Corollary 1.4 (0.51983 vs 0.53134) would be overwhelmed. However, this is not circularity - the asymmetric bound is an external input, not derived from the paper's own claims, and the main term of Theorem 1.3 does not depend on it. The house rules distinguish circularity (a reduction by construction) from correctness risk (an unverified transfer); this falls in the latter category. (5) There are no self-citations by the authors (references are to external works such as Katznelson, Lund, Huxley–Konyagin, Spencer, Thiele, Guy–Kelly), so patterns 3-5 do not apply. The computer-assisted constants in Lemmas 4.4/4.5 are stated with explicit error bounds and are verifiable in principle; there is no indication that the program was calibrated to force the 7/12 conclusion. Honest finding: no significant circularity

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

The central claims rest on standard or previously published results, not on new postulates. There are no fitted parameters. The computer-assisted constant γ is computed, not fitted.

assumptions (5)
  • domain assumption Katznelson's asymptotic count of integral matrices of fixed rank (Theorem 2.1)
    Used to derive Propositions 1.8 and 1.9, the edge counts for affine and linear degeneracies.
  • domain assumption Lund's bound on α-nondegenerate r-rich k-flat incidences (Lemma 3.3)
    Core tool in the upper bound for cospherical tuples S(n,d).
  • domain assumption Huxley-Konyagin's bound on asymmetric cyclic quadrilaterals (used in Lemma 4.2)
    Gives the O(n^{4+18/29+ε}) error term in the cyclic quadrilateral count.
  • standard math Spencer's deletion lemma (Lemma 1.7)
    Template that converts edge counts into independent-set lower bounds.
  • standard math Divisor bound d(m)=O(m^{c/log log m}) and Pick-style lattice polygon bound (Lemma 4.3)
    Used in Lemma 3.1 and Lemma 4.4.

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Pith. "Pith review of On subsets of lattice cubes avoiding affine and spherical degeneracies." pith.science (2026). https://pith.science/paper/RWDPBMSS

@misc{pith2026250906935,
  author       = {Pith},
  title        = {Pith review of: On subsets of lattice cubes avoiding affine and spherical degeneracies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RWDPBMSS}},
  note         = {Machine review of arXiv:2509.06935}
}
abstract

For integers $1 < k < d-1$ and $r \ge k+2$, we establish new lower bounds on the maximum number of points in $[n]^d$ such that no $r$ lie in a $k$-dimensional affine (or linear) subspace. These bounds improve on earlier results of Sudakov-Tomon and Lefmann. Further, we provide a randomised construction for the no-four-on-a-circle problem posed by Erd\H{o}s and Purdy, improving Thiele's bound. We also consider the random construction in higher dimensions, and improve the bound of Suk and White for $d \geq 4$. In each case, we apply the deletion method, using results from number theory and incidence geometry to solve the associated counting problems.

Figures

Figures reproduced from arXiv: 2509.06935 by the authors.

Figure 1
Figure 1. The shaded regions show P ′ ℓc across the three different regimes for c. We now compute the area of P ′ ℓc , which we shall use later to estimate mℓc . Let θ := tan−1 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. The sublattice L ⊂ (2Z) 2 of points (marked in black) which project onto integer points on the line ℓc (shown in green) across different regimes for a, b, c. Summing over all possible values of c, we conclude that the total number of isosceles trapezia with vertices in 2 · [m] 2 0 and their axis of symmetry in L, is equal to X 2bm c=−2am  mℓc 2  = 2 · X 2bm c=(b−a)m+1 " Area(P ′ ℓc ) 2 32(a 2 + b 2) 2 + 1 a 3 · O(… view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. No-$(k+1)$-in-line problem for $k \geqslant 3$

    math.CO 2026-07 accept novelty 8.0 of 10

    For k≥3 and sufficiently large n, the maximum number of points in an n×n grid with no k+1 collinear is exactly kn.

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