{"id":"78b96c0e-3bea-4c77-8d46-7393246a3c3e","arxiv_id":"2412.02866","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new construction gives subsets of the d-dimensional lattice cube of size n^{3/(d+1)-o(1)} with no d+2 points on a sphere or hyperplane, improving Thiele's 1995 bound.","lead":"For fixed dimension d, this paper constructs large sets of lattice points in the d-dimensional cube with no d+2 points lying on a common sphere or hyperplane. It improves the previous lower bound from n^{1/(d-1)} to n^{3/(d+1)-o(1)}.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The r-rich sphere bound in Section 3 silently replaces |S| by |S_r| without re-optimizing the partition parameter D; with D fixed, the resulting incidence bound is too weak and the dyadic sum no longer yields the claimed exponent.","rationale":"I agree with the reader's weakest-assumption analysis: the proof of Theorem 1.1 contains a genuine gap at the transition from the full sphere collection S to a dyadic subfamily S_r. The optimized incidence bound is obtained by balancing two terms through a parameter D that depends on |S|. For a proper subfamily, the same D no longer balances the corresponding two terms, and the first term can dominate the claimed |S_r|^{d^2/(d^2+d-1)} bound. I verified this numerically for d=3: with M=n^{12} and y=n^4, the fixed-D calculation gives O(n^6) whereas the claimed bound is n^{60/11}, so the asserted inequality is not justified. This is load-bearing because the subsequent dyadic summation relies crucially on the exponent d^2/(d^2+d-1) being strictly less than 1; with the weaker fixed-D bound the sum over dyadic scales would be too large by a polynomial factor. The gap seems repairable: one can run the incidence argument separately for each dyadic subfamily, choosing D_r as a function of |S_r|, and take a union bound over O(log n) possible dyadic D values so that the balance condition holds for A at every scale. But this repair is not present in the manuscript. I do not share the reader's secondary concern about Theorem 4.1: expanding the sphere equation along the moment curve, the coefficient of x^{2d-1} is indeed 0, so the sum of the 2d roots is 0 mod n; that part of the concluding remark is correct. This does not affect the main finding, which remains a conditional acceptance subject to fixing the S_r incidence step.","tokens_in":7854,"tokens_out":32317,"duration_ms":309026,"concrete_test":"Compute the fixed-D incidence bound for an arbitrary subfamily S'⊂S in Section 3. Take d=3, M=|S|=n^{12}, y=|S'|=n^4; the paper's choice gives D=Θ(1), hence I(S')≤C(n^3 y^{3/4}+y)=O(n^6). The claimed bound used in the S_r line is n^{24/11} y^{9/11}=n^{60/11}, which is smaller by n^{6/11}. If the authors instead re-derive the incidence bound with D_r=Θ(n^{12/11} y^{-1/11}) and show the required balance condition holds for A at that scale, the theorem is repairable; otherwise the proof step is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem hinges on the incidence estimate I(S_r) ≤ n^{3(d^2-1)/(d^2+d-1)+o(1)} |S_r|^{d^2/(d^2+d-1)} for every dyadic subfamily S_r. In the proof, however, this estimate is derived only for the full collection S, after choosing D so that n^{3(d+1)/(d^2+d-1)} |S|^{-1/(d^2+d-1)} ≤ D ≤ 2 n^{3(d+1)/(d^2+d-1)} |S|^{-1/(d^2+d-1)}. For a subfamily S_r with y=|S_r|, the same fixed D gives only I(S_r) ≤ n^{o(1)}( n^3 D^{-d^2/(d+1)} y^{d/(d+1)} + D^{d-1} y ). Substituting the D chosen for M=|S|, the second term is at most the claimed bound, but the first term can be larger. For d=3, M=n^{12}, y=n^4, D≈1, so the fixed-D bound is O(n^6), while the claimed bound is n^{60/11}=n^{5.45}; the former exceeds the latter by n^{6/11}. Thus the displayed step \"r|S_r| ≤ ... |S_r|^{d^2/(d^2+d-1)}\" does not follow from the preceding calculation. The dyadic summation needs the exponent d^2/(d^2+d-1) < 1 to converge; the weaker fixed-D bound would blow up the tuple count. The argument appears repairable by re-choosing D for each dyadic subfamily, provided the random set A is balanced at all O(log n) scales, but that re-optimization is absent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1577,"tokens_out":2139,"duration_ms":232617,"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":[{"comment":"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":"Section 3 (r-rich sphere estimate)"},{"comment":"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.","section":"Section 3 (multi-scale concentration)"}],"minor_comments":[{"comment":"The text says 'In this section, we prove Theorem 3' but the intended reference is Theorem 1.1.","section":"Section 3, first sentence"},{"comment":"'greatest common denominator' should read 'greatest common divisor'.","section":"Lemma 3.2"},{"comment":"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":"Lemma 2.2, Case 2"},{"comment":"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.","section":"Section 3, definition of S and the tuple count"},{"comment":"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.","section":"Deletion step, end of Section 3"}],"recommendation":"major_revision","confidential_remarks":"The main proof is not circular: Theorems 2.1 and 3.2 are external published results with independent proofs, and I saw no place where Theorem 1.1 is assumed. The single serious obstacle is the r-rich sphere estimate in Section 3, which needs a genuine new argument or a multi-scale concentration statement. I believe the underlying approach is likely repairable, but the missing step is central and cannot be waved away by a local rewording. If the authors supply a valid incidence bound for all dyadic subfamilies, the paper would be a solid contribution to the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nI 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.\n\nAll 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.","headline":"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.","tokens_in":8791,"tokens_out":6497,"would_cite":true,"duration_ms":62988,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52C10","05D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["no-d+2-on-a-sphere","lattice cube","spheres and hyperplanes","VC-dimension","incidence bounds","probabilistic method","no-four-on-a-circle","discrete geometry"],"falsifier":"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.","tokens_in":7616,"feed_emoji":"📐","tokens_out":9949,"duration_ms":87316,"temperature":0.7,"pith_summary":"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.","feed_headline":"Grid set of size $n^{3/(d+1)-o(1)}$ avoids d+2 points on a sphere","feed_subtitle":"Construction reaches exponent 3/(d+1) in all dimensions d≥3, beating the 1995 record of 1/(d−1).","key_machinery":"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 $I(S') \\le n^{O(1/\\log\\log n)}\\, n^{3(d^2-1)/(d^2+d-1)}\\, |S'|^{d^2/(d^2+d-1)}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the incidence bound for set systems with bounded shatter function that controls incidences in each subcube.","marker":"[7]"},{"why":"Supplies the bound on how many grid points can lie on a $(d-2)$-sphere, used to make the incidence graphs $K_{t,2}$-free.","marker":"[16]"},{"why":"Bounds the number of grid points on hyperplanes, used to control $(d+2)$-tuples on hyperplanes in the random set.","marker":"[2]"},{"why":"Gives the previous best lower bound $\\Omega(n^{1/(d-1)})$ that Theorem 1.1 improves.","marker":"[20]"},{"why":"Provides the shatter-function bound that converts the VC-dimension estimate into a polynomial shatter function.","marker":"[15]"},{"why":"Provides the concentration inequalities used to keep the random set and its subcube counts in the desired range.","marker":"[10]"}],"fun_headline_variants":["Grid set breaks 1995 record for sphere-avoidance in all dimensions","New bound n^(3/(d+1)-o(1)) avoids d+2 sphere points","Outperforming Thiele: sphere-free lattice sets with exponent 3/(d+1)","Lattice cube construction beats 1995 bound for no d+2 on a sphere"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Grid set breaks 1995 record for sphere-avoidance in all dimensions","New bound n^(3/(d+1)-o(1)) avoids d+2 sphere points","Outperforming Thiele: sphere-free lattice sets with exponent 3/(d+1)","Lattice cube construction beats 1995 bound for no d+2 on a sphere"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00056,"raw_usage":{"total_tokens":2612,"prompt_tokens":847,"completion_tokens":1765,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":1672}},"tokens_in":463,"tokens_out":1765,"duration_ms":12538,"temperature":1.0,"reasoning_tokens":1672,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:05:31.037214+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the incidence bound for set systems with bounded shatter function that controls incidences in each subcube."},{"cited_title":"Sheffer, Lower bounds for incidences with hypersurfac es, Discrete Analysis (2016), https://doi.org/10.19086/da912","cited_arxiv_id":null,"evidence_quote":"Supplies the bound on how many grid points can lie on a $(d-2)$-sphere, used to make the incidence graphs $K_{t,2}$-free."},{"cited_title":"Thiele, Geometric Selection Problems and Hypergraphs , Dissertation, FU Berlin 1995","cited_arxiv_id":null,"evidence_quote":"Gives the previous best lower bound $\\Omega(n^{1/(d-1)})$ that Theorem 1.1 improves."},{"cited_title":"Sauer, On the density of families of sets, J","cited_arxiv_id":null,"evidence_quote":"Provides the shatter-function bound that converts the VC-dimension estimate into a polynomial shatter function."},{"cited_title":"Janson, T","cited_arxiv_id":null,"evidence_quote":"Provides the concentration inequalities used to keep the random set and its subcube counts in the desired range."}],"review_version":1}