REVIEW 3 major objections 4 minor 2 cited by
Large grid subsets without many cospherical points
T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read This paper constructs, for every dimension d≥2, a subset of the grid [n]^d of size n−o(n) with no d+2 points on a common sphere or hyperplane.
desk verdict Strong new lower bound for no-d+2-cospherical grid subsets, sound at its core but with a few fixable gaps in the write-up. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the polynomial identity $f_1^2+\dots+f_d^2=gh$ over $F_p$, together with the degree data $(d,\ldots,d,d-1,d+1)$ and the linear independence of $f_1,\ldots,f_d,g,h$. This identity is what converts the sphere equation, quadratic in the coordinates, into a linear combination of $g$ and the coordinate functions after substitution along the curve; the result is a polynomial of degree at most $d+1$ whose roots are exactly the curve parameters lying on the sphere. The same substitution for a hyperplane gives a degree-$(d+1)$ polynomial. Projectively, the mechanism is that all spheres share the same points at infinity, and the choice of the $f_i$ makes the curve pass through enough of those points to cancel the quadratic contribution. For the refined coplanar statement, the extra 'nice' condition—zero linear coefficient on $f_1,\ldots,f_d$ and $h$—lets Vieta's formula forbid $d+1$ simultaneous zeros in the sampled reciprocal domain.
What would settle it
For $d=2$ and $p=13$, compute the curve from the paper's matrix with a square root $\alpha$ of $-1$, list the $10$ image points in $F_{13}^2$, and test every line and every sphere equation for four of those points; a single hit would refute the construction in that case, and since the theorem quantifies over all $d$, any such hit would disprove the claimed $n-o(n)$ bound.
Extended reading notes
Core claim
The central discovery is that a rational curve over a finite field can be engineered so that every sphere and every hyperplane pulls back to a low-degree polynomial, bounding the number of curve points on such a surface. The curve is $\gamma(t)=(f_1(t)/h(t),\ldots,f_d(t)/h(t))$ in $F_p^d$, where the polynomials satisfy $f_1^2+\dots+f_d^2=gh$. Substituting the curve into the equation of a sphere turns the quadratic part into $g$, leaving a polynomial of degree at most $d+1$; the same happens for a hyperplane. Because $f_1,\ldots,f_d,g,h$ are linearly independent and have degrees $(d,\ldots,d,d-1,d+1)$, neither pullback can vanish identically, and a nonzero degree-$(d+1)$ polynomial has at most $d+1$ roots. The curve has at most one self-intersection, so its image has $p-d-2$ distinct points, and a random translate of this image inside $[n]^d$, with the prime $p$ chosen just above $n$, yields the $n-o(n)$ subset. A refined condition on the linear coefficients lets the same curve be sampled on reciprocals, giving $n/(d+1)-o(n)$ points with no $d+1$ coplanar points.
Load-bearing premise
The construction collapses unless, for every dimension $d$ and infinitely many suitable primes, one can find the required polynomials over the finite field whose sum of squares factors as a product of two polynomials of the right degrees with the right independence; the paper constructs them explicitly, but it also assumes without proof that real spheres and planes behave the same way inside the finite field once a prime just above $n$ is chosen.
Editorial extensions
If this is right
- In the plane, the no-four-on-a-circle lower bound jumps from $n/4$ to $n-o(n)$, bringing it to within a constant factor of the trivial $3n$ upper bound.
- For every $d\ge 3$, the previously conjectured lower bound $n^{d/(d+1)}$ is confirmed in the much stronger linear form, so the extremal number is $\Theta(n)$ in each fixed dimension.
- The generalized grid problem is settled in growth rate: a linear lower bound of $n-o(n)$ matches the slicing upper bound $(d+1)n$ up to a sublinear factor.
- The construction works uniformly against both spheres and hyperplanes, so no separate handling of the two forbidden configurations is needed.
- The refined sampling on reciprocal parameters gives a positive-density subset with no $d+1$ coplanar points and no $d+2$ cospherical points, indicating the coplanar threshold can be pushed lower at the cost of density.
Reading between the lines
- A natural next step, not taken in the paper, would be to determine the leading constant: the true maximum lies somewhere between $n$ and $(d+1)n$, and the rational-curve construction alone does not say which side is closer.
- The same mechanism—a curve whose substitution makes a family of hypersurfaces pull back to low-degree polynomials—should apply to other configurations defined by quadrics that share points at infinity, such as ellipsoids, hyperboloids, or more general algebraic surfaces.
- Because the $o(n)$ error comes from the gap between the grid size $n$ and the chosen prime $p$, any improvement in the prime-selection step would immediately improve the error term; the current bound inherits the $n/\exp(c\sqrt{\log n})$ error from primes in arithmetic progressions.
- The reciprocal-sampling trick that forbids $d+1$ coplanar points suggests an analogy with no-three-in-line problems, where restricting parameter reciprocals may be a general way to avoid linear configurations; this is not explored in the paper.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the extremal function ex([n]^d; d+2), the largest subset of the d-dimensional grid with no d+2 points on a hyperplane or sphere. The main result, Theorem 2, claims ex([n]^d; d+2) >= n - o(n) for every d >= 2, which would asymptotically resolve the Brass-Moser-Pach problem and confirm the Suk-White conjecture in a strong form. The proof constructs an explicit rational curve over F_p whose image avoids d+2 coplanar and cospherical points, transfers it to the integer grid by a random translate, and uses a polynomial identity f_1^2+...+f_d^2=gh to control sphere and plane incidences. A secondary result, Theorem 3, claims ex([n]^d; d+1, d+2) >= n/(d+1)-o(n) using the same curve restricted to a smaller parameter set.
Significance. If the construction is fully correct, Theorem 2 is a striking improvement over the previous lower bounds of Thiele (n/4 for d=2) and Suk-White (Omega(n^{3/(d+1)-o(1)}) for d>=3), and it is tight up to the o(n) term against the trivial upper bound (d+1)n. The algebraic construction is elegant: the use of the identity f_1^2+...+f_d^2=gh is a genuinely new idea for controlling sphere incidences, and the self-intersection argument giving |S|>=p-d-2 is clean. The manuscript is self-contained in its polynomial construction and uses external results only as benchmarks or for prime distribution. The main concerns below are gaps in the written proof rather than in the overall strategy, and they appear repairable.
major comments (3)
- [Section 3, proof of Theorem 2] The sentence 'Since every plane (resp. sphere) in [n]^d is a plane (resp. sphere) in F_p^d' is false for spheres. A Euclidean sphere has a primitive integer equation A(x_1^2+...+x_d^2)+B_1x_1+...+B_dx_d+C=0. If p does not divide A, reducing modulo p gives an F_p sphere of the form used in the proof. But if p divides A, the reduction is a hyperplane (if some B_i or C is nonzero modulo p) or the empty equation (if all coefficients are 0 modulo p, which cannot happen for a primitive equation). In the hyperplane case the already-proved no-(d+2)-coplanar property of S in F_p applies, and in the empty case there are no grid points at all. This case analysis is missing, so the transfer step is not justified as written; the claim is salvageable exactly as described, but it is load-bearing for Theorem 2.
- [Section 2, paragraph after Eq. (1)] The proof asserts that after the replacements f_i -> f_i+mu_i g and f_1 -> f_1+(mu_1+nu t)g, 'identity, degree, and independence are left intact' for the later chosen nu. This is not automatic. Since g is monic of degree d-1 and f_1 has leading coefficient 1, the coefficient of t^d in f_1+(mu_1+nu t)g is 1+nu, so the value nu=-1 would drop the degree of f_1. More generally, if t g lies in the span of f_1,...,f_d,g, the linear transformation from (f_1,...,f_d,g) to the new polynomials has determinant 1+nu c_1 for some c_1, so it can become singular for a specific nu. The proof must show that the unique nu making [t]h tilde =0 avoids these exceptional values, or must prove independence of the final family by a separate argument. Because the incidence proofs in Theorems 2 and 3 use the degree and independence bullets of Theorem 4, this gap is load-bearing.
- [Section 3, proof of Theorem 3] The Vieta step is not justified. The proof says that if P has d+1 distinct zeros t_1,...,t_{d+1} in the domain, then t_1^{-1}+...+t_{d+1}^{-1}=0 in F_p, and 'This implies that such zeros cannot live in the domain of e-gamma simultaneously.' As a modular statement this is false: for d=2 and p=13 the domain is {2,3,4,5}, and 2^{-1}+3^{-1}+4^{-1}=7+9+10=26=0 mod 13. Thus the argument needs an additional ingredient, for example a property specific to the span of f_1,...,f_d,h, or a different choice of parameter set, to rule out d+1 zeros. This gap affects Theorem 3 rather than Theorem 2, but Theorem 3 is a stated result of the paper.
minor comments (4)
- [Section 3, proof of Theorem 3] In the sentence 'If P has d+1 distinct zeros t_1,...,t_{d+1} in {1,2,...,n}', the symbol n is undefined and appears to be a typo; the zeros should range over the domain of e-gamma, which is a subset of F_p.
- [Section 3, proof of Theorem 3] The phrase 'the image of gamma' in the proof of Theorem 3 should refer to the image of e-gamma; as printed it repeats the notation from Theorem 2.
- [Section 2, proof of Claim 6] In Claim 6, the displayed equation 'c_1a_{1,j}+...+c_da_{d,j} holds' is missing '=0'; the intended conclusion is that the row vectors of A are linearly dependent.
- [Section 2, after the proof of Theorem 4] There is a typo 'Since h is is of higher degree' with a duplicated 'is'; also in the self-intersection paragraph of the proof of Theorem 2 the polynomial R is written as 'rho_1f_1+...+rho_df_d+rho f' but should be 'rho h'.
Circularity Check
No circularity: the construction is explicit and self-contained, with prior work cited only as context and comparison.
full rationale
The central derivation is carried out within the paper rather than imported from a prior result or inferred from the target extremal quantity. Theorem 4 constructs the polynomials f1,...,fd,g,h explicitly: it fixes distinct roots λ1,...,λd, defines a circulant-style matrix A, proves linear independence by evaluation at the λj, proves that g divides the sum of squares so that h is defined as the quotient, and then enforces the additional 'nice' condition by explicit perturbations with parameters µi and ν. The identity f1^2+...+fd^2=gh is therefore an algebraic consequence of the construction, not an assumed input. Theorem 2 then uses only degree, independence, and this identity to bound intersections of the parametric curve with arbitrary planes and spheres in F_p^d; the argument counts roots, so the bound is deterministic and not statistically fitted. There are no fitted parameters, no calibration to external bounds, and no subset of the target data used to produce a 'prediction.' Cited works such as Thiele, Suk-White, and Brass-Moser-Pach are used to define the problem and to state the previous bounds, not to supply any load-bearing ingredient. The only external theorem used is the standard prime number theorem in arithmetic progressions, which is independent and commonly available. The paper is therefore self-contained with respect to its own claimed derivation. To be clear, the assertion that every Euclidean sphere or plane in [n]^d is a sphere or plane in F_p^d is stated without a full case analysis, and the verification of the column sum-of-squares condition in Claim 7 deserves a careful check; these are potential correctness issues rather than circular reductions. Neither relies on assuming Theorem 2 or the extremal function ex([n]^d; d+2).
Assumptions & free parameters
assumptions (5)
- standard math Prime number theorem for arithmetic progressions (Theorem 9)
- standard math Finite field F_p with p ≡ 1 (mod 4) contains a square root of -1
- domain assumption Any d points in F_p^d lie on a hyperplane
- standard math Vieta's theorem for polynomials over F_p
- domain assumption Identification of Euclidean spheres and planes in [n]^d with F_p spheres and planes after reduction mod p
Cite this review
Pith. "Pith review of Large grid subsets without many cospherical points." pith.science (2026). https://pith.science/paper/4NU327MU
@misc{pith2026250618113,
author = {Pith},
title = {Pith review of: Large grid subsets without many cospherical points},
year = {2026},
howpublished = {\url{https://pith.science/paper/4NU327MU}},
note = {Machine review of arXiv:2506.18113}
}
abstract
Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n]^d$ of size $n - o(n)$ with no $d + 2$ points on a sphere or a hyperplane. For $d = 2$, this improves the previously best known lower bound of $n/4$ toward the Erd\H{o}s--Purdy problem due to Thiele in 1995. For $d \ge 3$, this improves the recent $\Omega \bigl( n^{\frac{3}{d+1}-o(1)} \bigr)$ bound due to Suk and White, confirming their conjectured $\Omega \bigl( n^{\frac{d}{d+1}} \bigr)$ bound in a strong sense, and asymptotically resolves the generalized Erd\H{o}s--Purdy problem posed by Brass, Moser, and Pach.
Forward citations
Cited by 2 Pith papers
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No-$(k+1)$-in-line problem for $k \geqslant 3$
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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On subsets of lattice cubes avoiding affine and spherical degeneracies
New lower bounds for lattice sets avoiding subspheres and subspaces, including f_circ(n) ≥ 7n/12, via deletion-method counting of cyclic quadrilaterals and cospherical tuples.
Reference graph
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Reviewed August 15, 2026 · model on record in the stance chip above.
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