A note on the no-$(d+2)$-on-a-sphere problem

Andrew Suk; Ethan Patrick White

REVIEW 2 major objections 5 minor 1 cited by

A note on the no-$(d+2)$-on-a-sphere problem

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For each d≥3, a d-dimensional grid set of size $n^{3/(d+1)-o(1)}$ avoids d+2 points on any sphere or hyperplane.

desk verdict New lower-bound construction for the no-(d+2)-on-a-sphere problem that likely beats Thiele, but the main incidence estimate has a real gap: the partition parameter D is not re-optimized when passing to dyadic subfamilies. read the letter →

arxiv 2412.02866 v1 pith:QWYMVOUG submitted 2024-12-03 math.CO cs.DM

Andrew Suk , Ethan Patrick White This is my paper · ORCID

classification math.COcs.DM MSC 52C1005D40

keywords no-d+2-on-a-spherelatticecubespheresandhyperplanesVC-dimensionincidenceboundsprobabilisticmethodno-four-on-a-circlediscretegeometry

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 proves Theorem 1.1: for every fixed $d\ge 3$, some subset of the lattice cube $[n]^d$ has $n^{3/(d+1)-o(1)}$ points with no $d+2$ of them lying on a common sphere or hyperplane. This improves the previous best lower bound, $\Omega(n^{1/(d-1)})$ from 1995, and it is the strongest known construction for the no-$(d+2)$-on-a-sphere problem in fixed dimension. The construction is probabilistic: a moderately dense random subset is selected, the number of rich spheres and hyperplanes it contains is bounded with incidence geometry, and one point is deleted from every forbidden $(d+2)$-tuple. A sympathetic reader would care because the result nearly closes the gap to the paper's conjectured bound $n^{d/(d+1)}$ and shows that VC-dimension incidence bounds can control sphere-rich lattice sets.

What carries the argument

The load-bearing mechanism is the incidence bound between the random set $A$ and the family $\mathcal{S}$ of spheres that contain at least $d+1$ points of $A$ in general position. The proof partitions the cube into $D^d$ subcubes, uses the fact that the intersection of two spheres is a $(d-2)$-sphere with small grid content, and applies an incidence theorem for set systems with bounded shatter function to each incidence graph $G_j=(P_j,S_j,E_j)$. The VC-dimension of the family of maximal spherical sets---the largest point set whose subsets can all be realized as intersections with spheres---is at most $d+1$, so the shatter function is $O(z^{d+1})$; this feeds the incidence bound, and choosing $D$ balances two terms to yield $I(S') \le n^{O(1/\log\log n)}\, n^{3(d^2-1)/(d^2+d-1)}\, |S'|^{d^2/(d^2+d-1)}$.

What would settle it

For a fixed dimension $d$ and a dyadic subfamily $S_r$ of spheres, compute both sides of the displayed incidence inequality with the partition parameter $D$ fixed at the value chosen for the full family $\mathcal{S}$; if the inequality fails for small $|S_r|$, then the proof's use of a single $D$ for all subfamilies would need to be modified.

Watch

Extended reading notes

Core claim

The paper establishes that, for fixed $d\ge 3$, there exists a set $A\subset [n]^d$ with $|A| = n^{3/(d+1)-o(1)}$ such that no $d+2$ members of $A$ lie on a $(d-1)$-sphere or on a hyperplane. The proof selects points independently with probability $n^{3-d}$, partitions $[n]^d$ into $D^d$ equal subcubes, and applies an incidence theorem for set systems of bounded shatter function to each subcube's sphere-incidence graph. After balancing the partition parameter $D$, the authors obtain an upper bound of order $n^{3(d+1)+o(1)}$ on the number of $(d+2)$-tuples lying on a common sphere, and then a random deletion step removes one point from each bad tuple. The final subset retains at least $n^{3/(d+1)-o(1)}$ points with no forbidden configuration.

Load-bearing premise

The proof assumes that the incidence bound proved for the full family of spheres keeps the same strength when applied to a smaller subfamily, even though the partition scale that optimizes the bound could depend on the size of that subfamily.

Editorial extensions

If this is right

The best known lower bound for the no-$(d+2)$-on-a-sphere problem in fixed dimension $d\ge 3$ becomes $n^{3/(d+1)-o(1)}$, improving the 1995 bound $\Omega(n^{1/(d-1)})$.
Since any hyperplane contains at most $d+1$ points and $[n]^d$ is covered by $n$ hyperplanes, the upper bound remains $(d+1)n$, so the gap between upper and lower exponents shrinks from roughly $1-1/(d-1)$ to $1-3/(d+1)$.
The same random-selection-plus-deletion template yields $n^{2-4/(d+1)-o(1)}$ points with no $d+2$ on a sphere when many points on hyperplanes are allowed.
The modular moment-curve construction in Theorem 4.1 gives an independent, deterministic $\Omega(n)$-size set with no $d+1$ points on a hyperplane and no $2d$ points on a sphere.
The paper's Conjecture 1.2, proposing an $\Omega(n^{d/(d+1)})$ lower bound, becomes the natural next step beyond the achieved exponent.

Reading between the lines

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

If the sphere-incidence exponent could be sharpened within the same VC-dimension template, the proof would likely push the lower bound toward the conjectured $n^{d/(d+1)}$ exponent.
The proof's dependence on the $(d-2)$-sphere intersection property suggests that a similar construction may yield large sphere-free subsets for other forbidden tuple sizes by replacing $d+2$ with a general $k$ and tracking the VC-dimension parameter.
The fixed-partition-scale step could be tested numerically on small $d$: recomputing the incidence bound with a partition scale chosen for each subfamily of spheres would show whether the stated exponent survives for sparse subfamilies.
Combining the modular moment-curve idea with the deletion method might give deterministic or explicit constructions with the same $n^{3/(d+1)-o(1)}$ exponent, since the arithmetic sum-of-roots obstruction already controls many sphere incidences.

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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

Here's my read of Suk-White. The main result is a genuine improvement: n^{3/(d+1)-o(1)} beats Thiele's n^{1/(d-1)} for d≥3, and the construction is clean—random n^3-point set, partition into D^d boxes, semi-algebraic Zarankiewicz box by box, then deletion. The VC-dimension and K_{t,2}-freeness setup is sound, and the overall strategy is worth stealing. The paper is short and readable, and the citations check out as legitimate external theorems, including the ones with overlapping authorship.

The soft spot is exactly where your reader put it. The step r|S_r| ≤ n^{...}|S_r|^{...} is not justified as written. The incidence bound is derived after choosing D as a function of |S|. When you pass to S_r, that same D is generally too coarse, so the first term n^3 D^{-d/(d+1)} y^{d/(d+1)} dominates and the resulting bound is too weak. For d=3, with |S| ~ n^{12} and y = n^4, the fixed-D bound is O(n^6) rather than n^{60/11}. The standard repair is to re-choose D for each dyadic subfamily, which needs the random A to be balanced for all D in a range of O(log n) values. That can be done by a union bound and probably preserves the claimed exponent, but it is not in the paper. So the main theorem is conditional, not established.

I do disagree with one part of the reader's report. The Theorem 4.1 coefficient complaint is not a real error: the coefficient of x^{2d-1} in the sphere polynomial (x-c_1)^2 + ... + (x^d-c_d)^2 - r^2 is zero, so the sum of the roots is 0 mod n. The division-algorithm argument is fine. The real issue stays in Section 3.

All in all: the core idea is solid and the failure is a technical but load-bearing gap in the proof. This deserves a serious referee, because the construction is new, the exponent is meaningful, and the gap is almost certainly repairable. I'd treat it as major-revision material, not a desk reject.

Referee Report

2 major / 5 minor

Summary. The paper studies the higher-dimensional analogue of the no-four-on-a-circle problem, asking how many points of the lattice cube [n]^d can be selected with no d+2 points on a (d-1)-sphere or hyperplane. The main result, Theorem 1.1, constructs a subset of size n^{3/(d+1)-o(1)} with this property for every fixed d>=3, improving the previous n^{1/(d-1)} lower bound of Thiele. The proof is probabilistic: a random subset A of [n]^d is drawn with density n^{3-d}, several concentration properties are verified, the collection S of spheres determined by general-position (d+1)-tuples of A is partitioned into cells, an incidence bound from VC-dimension theory is applied to each cell, and dyadic counting of r-rich spheres leads to an estimate on the number of forbidden (d+2)-tuples, which is then removed by random deletion. The paper also contains a short elementary construction (Theorem 4.1) giving Omega(n) points with no d+1 on a hyperplane and no 2d points on a sphere.

Significance. If Theorem 1.1 is correct, it is a substantial polynomial improvement over the previous best bound for all d>=3, and the proof introduces a modern incidence-geometric framework into this classical lattice problem. The use of the Fox-Pach-Sheffer-Suk-Zahl incidence bound and of the Balogh-White lattice-point estimate is natural and likely to be useful beyond this specific question. The claimed exponent n^{3/(d+1)-o(1)} is plausible, and the paper is clearly written. However, the central r-rich sphere estimate contains a derivation gap that is load-bearing for the main theorem; the manuscript is therefore not yet in publishable form.

major comments (2)

[Section 3 (r-rich sphere estimate)] The step immediately after defining S_r is not justified. The preceding incidence bound for S was established only after choosing the partition parameter D as a function of the full collection size |S|, namely D approximately n^{3(d+1)/(d^2+d-1)} |S|^{-1/(d^2+d-1)}. The displayed inequality r|S_r| <= n^{3(d^2-1)/(d^2+d-1)+c_8/log log n} |S_r|^{d^2/(d^2+d-1)} silently replaces |S| by |S_r| in that bound. With the same fixed D, the first term of the incidence estimate for a subfamily S_r of size y=|S_r| is O(n^{3+c/log log n} D^{-d/(d+1)} y^{d/(d+1)}), and substituting the D chosen for |S| gives O(n^{3(d^2-1)/(d^2+d-1)+c/log log n} (|S|/y)^{d/((d+1)(d^2+d-1))} y^{d^2/(d^2+d-1)}). For y<|S| this exceeds the claimed bound by a polynomial factor; for example, when d=3, |A|=n^3, |S|=n^{12}, and y=n^4, the extra factor is n^{6/11}. The dyadic summation requires the exponent d^2/(d^2+d-1) on |S_r| to converge, and the fixed-D bound does not provide that exponent. This is a load-bearing gap: the final tuple count and the deletion argument depend on this dyadic estimate.
[Section 3 (multi-scale concentration)] A repair by re-optimizing D separately for each dyadic subfamily S_r is not written, and it is not a purely cosmetic change. Property 2 of the event W is proved for one fixed D chosen before A is drawn; the proof as written does not establish concentration of |Q_j intersect A| for the many different D values that would be needed if each S_r used its own optimal D. In addition, for small |S_r| the optimizing D_r may exceed n, in which case the partition argument with the stated concentration property is not applicable at all. The manuscript needs either a simultaneous concentration statement over all relevant scales or a different argument handling the range where the optimal D is not admissible.

minor comments (5)

[Section 3, first sentence] The text says 'In this section, we prove Theorem 3' but the intended reference is Theorem 1.1.
[Lemma 3.2] 'greatest common denominator' should read 'greatest common divisor'.
[Lemma 2.2, Case 2] In the induction step, the statement that Q cannot be shattered by S_{n,d} is imprecise: the shattering by spheres restricted to the hyperplane h is governed by the (d-2)-sphere set system on h, i.e., by the induction hypothesis for dimension d-1 rather than by S_{n,d} itself.
[Section 3, definition of S and the tuple count] The proof counts (d+2)-tuples on spheres S in S, but S was defined to consist only of spheres containing at least d+1 points of A in general position. A (d+2)-tuple lying on a sphere but with no such general-position subset is necessarily contained in a hyperplane and is therefore handled by the separate hyperplane bound; this reduction should be stated explicitly.
[Deletion step, end of Section 3] The passage 'By Lemma 3.1, there is a subset A' of A of size at least p|A|/2...' jumps from Chernoff's inequality to a joint statement about size and number of bad tuples; a brief union-bound or probabilistic-method sentence would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the target bound is never assumed; the cited incidence and lattice-point theorems are external results and do not encode the conclusion.

full rationale

The proof of Theorem 1.1 constructs a random set A with probability n^{3-d}, enforces four high-probability properties using Chernoff, Sheffer's sphere-point bound, and the Balogh--White hyperplane-point bound, and then counts incidences between A and spheres via the Fox--Pach--Sheffer--Suk--Zahl Zarankiewicz theorem. The target no-(d+2)-on-a-sphere bound is not an input to any of these tools; it is derived only after the dyadic r-rich sphere count and the final deletion argument. The cited results are independent prior theorems: [7] is a published JEMS theorem with its own proof, [16] is a published Discrete Analysis result, and [2] is a separate Balogh--White lemma on lattice hyperplanes. Although two citations have author overlap with the present note, none of them assumes, restates, or is equivalent to Theorem 1.1, so under the stated rules these citations are real evidence rather than circular support. The parameter D is chosen by balancing the incidence bound, and the final exponent n^{3/(d+1)-o(1)} is the output of that optimization, not a renaming of the input. The one substantive concern is internal rather than circular: after D is fixed using the full collection S, the displayed step 'r|S_r| <= n^{...}|S_r|^{...}' applies the same bound to each dyadic subfamily without re-optimizing D for |S_r|, so the dyadic estimate may not follow as written. A repair would re-choose D per subfamily using the same independent incidence theorem; the missing argument is therefore a proof gap, not an assumption of the target result.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof introduces no fitted constants or new entities. It relies on several external theorems, some co-authored by the current authors, but they are established results with independent proofs. The gap in the S_r incidence step is not an axiom but a missing derivation.

assumptions (6)

standard math Sauer-Shelah lemma: if a set system has VC-dimension d0 then its primal shatter function is bounded by the sum of binomial coefficients.
Used in Section 3 to bound shatter functions of sphere families.
standard math Chernoff bound for binomial random variables.
Used to show the random set A satisfies the desired properties with high probability.
domain assumption Sheffer's bound (Lemma 3.4): every k-sphere in R^d contains at most n^{k-1+c/log log n} points from [n]^d.
Used in property 3 of the random set A; a non-trivial external result.
domain assumption Balogh-White bound (Lemma 3.2): at most 3d n^{d-1}/s points of [n]^d lie on a primitive hyperplane with normal max coefficient s.
Used to count (d+2)-tuples on hyperplanes.
domain assumption Fox-Pach-Sheffer-Suk-Zahl theorem (Theorem 2.1): Zarankiewicz bound for semi-algebraic bipartite graphs with bounded VC-dimension.
Central tool for bounding sphere-point incidences inside each subcube.
standard math Turan's theorem: any graph with max degree Δ has an independent set of size at least |V|/(Δ+1).
Used in Observation 3.5 to extract a separated subset of crossed subcubes.

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Cite this review

Pith. "Pith review of A note on the no-$(d+2)$-on-a-sphere problem." pith.science (2026). https://pith.science/paper/QWYMVOUG

@misc{pith2026241202866,
  author       = {Pith},
  title        = {Pith review of: A note on the no-$(d+2)$-on-a-sphere problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QWYMVOUG}},
  note         = {Machine review of arXiv:2412.02866}
}

abstract

For fixed $d\geq 3$, we construct subsets of the $d$-dimensional lattice cube $[n]^d$ of size $n^{\frac{3}{d + 1} - o(1)}$ with no $d+2$ points on a sphere or a hyperplane. This improves the previously best known bound of $\Omega(n^{\frac{1}{d-1}})$ due to Thiele from 1995.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

On subsets of lattice cubes avoiding affine and spherical degeneracies
math.CO 2025-09 conditional novelty 6.0 of 10

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

Works this paper leans on

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Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

N. Alon, J. Spencer, The Probabilistic Method, 3rd edn. W iley Interscience (2008)

work page 2008

[2] [2]

Balogh, E

J. Balogh, E. P. White, Grid drawings of graphs in three-dimensions , arxiv preprint , arXiv:2404.02369 (2024)

arXiv 2024

[3] [3]

Braß and C

P. Braß and C. Knauer, On counting point-hyperplane inci dences, Computational Geometry 25 (2003), 13–20

work page 2003

[4] [4]

Brass, W

P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry . Berlin, Germany: Springer-Verlag, 2005

work page 2005

[5] [5]

Dudeney, Amusements in Mathematics , Nelson, London, 1917

H.E. Dudeney, Amusements in Mathematics , Nelson, London, 1917

work page 1917

[6] [6]

Flammenkamp, Progress in the no-three-in-line probl em, II, Journal of Combinatorial The- ory, Series A 81 (1998), 108–113

A. Flammenkamp, Progress in the no-three-in-line probl em, II, Journal of Combinatorial The- ory, Series A 81 (1998), 108–113

work page 1998

[7] [7]

J. Fox, J. Pach, A. Sheﬀer, A. Suk, J. Zahl, A semi-algebrai c version of Zarankiewicz’s problem, J. Eur. Math. Soc. 19 (2017), 1785–1810

work page 2017

[8] [8]

R. Hall, T. Jackson, A. Sudbery, and K. Wild, Some advance s in the no-three-in-line problem, Journal of Combinatorial Theory, Series A 18 (1975), 336–341

work page 1975