{"id":"e1a0feaa-6a41-4137-aa63-97842194c24d","arxiv_id":"2505.23335","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A corrected version of Costello's conjecture holds for multilinear polynomials with optimal exponent 1, complex quadratics get a 13/24 power saving, and the original conjecture is false for degree at least 3.","lead":"The paper proves that polynomials of independent random signs can only concentrate on a single value with unusually high probability when they are close to an algebraically special form, and it shows a 2013 conjecture of this kind is false. It provides optimal power-saving bounds in the multilinear case and a partial quadratic case, using new local-to-global lemmas that also give property testers for low-rank tensors.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"False disjoint-bases estimate in Appendix A's proof of Theorem 2.5 invalidates the n^{-1/2} exponent used in Theorem 1.9.","rationale":"The paper's flagship theorem (Theorem 1.9) is proved by induction on d, and the induction step relies on Theorem 2.4, whose proof applies the geometric Littlewood-Offord theorem (Theorem 2.5) to the variety of reducible tensors. Theorem 2.5 is stated with an n^{-1/2+ε} bound unless a translate of a subspace contained in Z captures almost all vectors. The appendix purports to derive Theorem 2.5 from the Meka-Nguyen-Vu polynomial Littlewood-Offord theorem. The derivation hinges on constructing Θ(n) disjoint bases of a b-dimensional subspace W. The claimed lower bound (n_b−n_{b−1})/b ≥ ε n/d^2 is not implied by the minimality of b: with n_b = n−ε n^{(d−b)/d}, the difference n_b−n_{b−1} is Θ(n^{(d−b+1)/d}), which for b=d is Θ(n^{1/d}), not Θ(n). A concrete configuration (n−√n points on one axis, √n on another, with Z={0}) satisfies the minimality condition with b=d=2 but yields only Θ(√n) disjoint bases. Consequently the proof's m = Θ(n) cannot be realized, and the constructed polynomial need not have Ω(n^q) nonzero coefficients; the resulting bound would be n^{−1/(2d)+o(1)}, which is insufficient for Theorem 2.4's D2 and the induction in Theorem 6.1. Because the theorem is cited to [18], the mathematical result may still be correct, but the manuscript's self-contained argument contains a real gap. The reader's verdict should therefore be CONDITIONAL pending a corrected proof of Theorem 2.5 or an explicit reliance on the external reference. This is a different concern from the reader's flagged Nguyen-Vu inverse theorem, which applies only to Theorems 1.10 and 1.11, not to the central Theorem 1.9.","tokens_in":56,"tokens_out":33787,"duration_ms":853156,"concrete_test":"Specialize the Appendix A proof to d=2, F=R, Z={0}. For n=10^6 and ε=0.1, let the vectors a_i consist of n−√n copies of (1,0) and √n copies of (0,1). Check that no line contains n_1 = n − ε√n vectors, so b=2, and that the number of disjoint bases of F^2 is √n = 1000, while the asserted lower bound is ε n/d^2 = 25000. This disproves the key inequality; then verify whether the remainder of the proof can still produce a polynomial with Ω(n) nonzero coefficients for this configuration.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Theorem 2.5 (Appendix A), the authors define n_b = n − ε n^{(d−b)/d} and let b be minimal such that at least n_b of the vectors a_i lie in a common b-dimensional subspace W. They then claim that minimality yields at least (n_b − n_{b−1})/b ≥ ε n/d^2 disjoint bases of W. This is false for b = d: n_d − n_{d−1} = ε(n^{1/d} − 1), so the number of disjoint bases is Θ(n^{1/d}), not Θ(n). For example, when d=2, Z={0}, and the vectors consist of n−√n copies of (1,0) and √n copies of (0,1), no line contains n−ε√n vectors (so b=2), yet only √n disjoint bases of F^2 exist, far fewer than ε n/4. The proof then sets m = ⌊ε n/d^2⌋ b and relabels the first m vectors as disjoint bases; without Θ(n) bases, the resulting polynomial has only n^{q/d} nonzero coefficients, giving a bound of n^{−1/(2d)+o(1)} instead of n^{−1/2+ε}. Since Theorem 2.5 is used in Theorem 2.4 and hence in the induction proving Theorem 1.9, the central n^{−1} claim for multilinear forms is not justified by the paper's own proof. The theorem is credited to [18], so a correct proof may exist, but the manuscript as written contains this gap.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies anticoncentration of degree-d polynomials of independent Rademacher variables, in the direction of the polynomial Littlewood–Offord problem. Its main results are: an optimal n^{-1+o(1)} point-probability bound for d-multilinear forms that are far from reducible (Theorem 1.9), a power-saving bound n^{-1/2-1/24+o(1)} for robustly irreducible complex quadratic polynomials (Theorem 1.10), and a rank-dependent bound n^{-1+2/k+o(1)} for quadratic polynomials whose quadratic part has high rank (Theorem 1.11). The paper also contains an appendix disproving Costello's original conjecture on multilinear forms, together with local-to-global lemmas for tensor reducibility and symmetric matrix rank that are of independent interest for property testing. The proofs are organized around explicit epsilon-delta parameters and are mostly self-contained, but the proof of the geometric Littlewood–Offord theorem in Appendix A contains a serious gap that affects the proof of Theorem 1.9.","tokens_in":58329,"tokens_out":8883,"duration_ms":107745,"significance":"If the results are correct, Theorem 1.9 essentially resolves the repaired multilinear version of Costello's conjecture at the optimal n^{-1} exponent, since examples from the multiplication-table problem show that no bound better than (log n)^O(1)/n is possible. Theorem 1.10 gives the first power-saving beyond the Meka–Nguyen–Vu bound under an algebraic irreducibility assumption in the complex quadratic case, and Theorem 1.11 provides a partial answer to the rank-based version of the problem. The local-to-global lemmas for tensors and symmetric matrices, and the appendix disproof of Costello's original conjecture, are substantial contributions in their own right. The paper is also commendable for stating all quantitative parameters explicitly rather than hiding them in unspecified constants. However, the central multilinear theorem relies on a flawed proof in Appendix A, so the paper is not yet in publishable form.","major_comments":[{"comment":"The disjoint-bases claim in the proof of Theorem 2.5 is incorrect. The proof defines n_b = n - \\epsilon n^{(d-b)/d}, lets b be minimal such that at least n_b of the vectors a_i lie in a common b-dimensional subspace W, and then asserts that minimality yields at least (n_b - n_{b-1})/b \\ge \\epsilon n/d^2 disjoint bases of W. For b=d this is false: n_d - n_{d-1} = \\epsilon(n^{1/d}-1), so the number of disjoint bases is only about \\epsilon n^{1/d}/d, not linear in n. A concrete example in d=2 is obtained by taking n-\\sqrt{n} copies of (1,0) and \\sqrt{n} copies of (0,1) with Z={0}; then no line contains n-\\epsilon\\sqrt{n} vectors, so b=2, yet only \\sqrt{n} disjoint bases exist. Consequently the later choice m = \\lfloor \\epsilon n/d^2 \\rfloor b is not legitimate, and the coefficient-counting argument yields only m^q = n^{q/d} nonzero coefficients rather than n^q. The resulting bound is n^{-1/(2d)+o(1)} instead of the claimed n^{-1/2+\\epsilon}.","section":"Appendix A, proof of Theorem 2.5"},{"comment":"The gap in Appendix A propagates to the proof of the paper's main multilinear theorem. Theorem 2.5 is used in the proof of Lemma 2.4 to obtain the n^{-1/2+\\epsilon} bound in D2, and Lemma 2.4 is then an essential induction step in the proof of Theorem 6.1, which implies Theorem 1.9. Since the proof of Theorem 2.5 presented in the manuscript does not establish the claimed exponent, the proof of Theorem 1.9 is incomplete as written. The authors should either repair the disjoint-bases argument, or explicitly state that Theorem 2.5 is imported from the published paper [18] and remove the flawed self-contained proof.","section":"Section 6, proof of Theorem 6.1 via Theorem 2.4"}],"minor_comments":[{"comment":"The text frequently switches between calling the same statement a Lemma and a Theorem (for example, Lemma 2.2, Lemma 2.4, and Lemma 2.7 are sometimes referred to as Theorems 2.2, 2.4, and 2.7). These labels should be made consistent.","section":"Throughout"},{"comment":"In the proof of Theorem 6.1, the event E_1 is defined using epsilon_1 = epsilon^3/32 and the bound |I'|^{-1/2+epsilon_1} is stated as <= n^{-1/2+epsilon/2}; since epsilon_1 is much smaller than epsilon/2 this is fine, but the display would be clearer if the inequality were written as |I'|^{-1/2+epsilon_1} <= n^{-1/2+epsilon/2} after using |I'| >= \\epsilon n.","section":"Section 6"},{"comment":"After the proof of Claim B.5, the text says 'Theorem B.5 tells us...' but the statement is a claim, not a theorem; similarly, several equations in Section 10 refer to 'Theorem 10.2' and 'Theorem 10.5' where 'Lemma 10.2' and 'Lemma 10.5' would be more accurate. These numbering inconsistencies should be cleaned up.","section":"Appendix B, Claim B.5"},{"comment":"In the proof of Theorem 2.17, the exponent calculations in the estimates for the second and third terms are terse; for example, the step from n^{-3/2+5\\delta} E[...] to n^{-2-1/6+7\\delta} uses Theorem 10.6, but it would help the reader if the inequalities were displayed separately rather than compressed into one chain.","section":"Section 10.1"}],"recommendation":"major_revision","confidential_remarks":"The paper contains strong and interesting results, and the appendix disproof of Costello's conjecture is a valuable contribution. However, the proof of Theorem 2.5 in Appendix A has a concrete and load-bearing error: the disjoint-bases count is wrong when b=d, and this is exactly the case needed for the proof of Theorem 1.9. Since Theorem 2.5 is credited to [18], the authors can likely repair the paper by citing the published result and removing or correcting the flawed proof, so I do not recommend rejection; but the manuscript should not be accepted with the current gap. I would also ask the authors to double-check whether the same disjoint-bases argument appears in [18]; if it does, the flaw may indicate a typographical error in their exposition rather than in the underlying theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers genuinely new results: the corrected multilinear Costello conjecture with exponent 1 (Theorem 1.9), the complex quadratic power saving 13/24 (Theorem 1.10), the high-rank quadratic bound approaching 1/n (Theorem 1.11), and the disproof of Costello's original conjecture in the appendix. The local-to-global lemmas for tensors and symmetric matrices are nice and should have independent life in property testing. The counterexample construction is clever and well worth having.\n\nThe soft spot is in Appendix A, and the stress-test note is right. In the proof of Theorem 2.5, after choosing b minimal with n_b vectors in a b-dimensional subspace W, the paper claims minimality yields at least (n_b - n_{b-1})/b >= epsilon n/d^2 disjoint bases of W. For b=d, n_d - n_{d-1} = epsilon(n^{1/d}-1), so this is false; the correct lower bound is only about n^{1/d}, not n. The example with d=2, many copies of (1,0) and few of (0,1) shows the issue. This invalidates the proof of Theorem 2.5 as written, and since Theorem 2.5 is used to prove Theorem 2.4 and then Theorem 1.9, the self-contained proof chain has a genuine gap. The theorem is credited to Fox-Kwan-Spink [18], so the main results may be recoverable by simply citing that work and removing the faulty appendix proof, but the manuscript as written does not provide a correct proof.\n\nThe rest of the paper looks solid: the dependence on the Nguyen-Vu inverse theorem is a reasonable black box, no fitted constants or circular normalizations, and the induction for Theorem 1.9 is otherwise coherent. The reliance on deep imported results is worth flagging but does not rise to an error.\n\nThis paper deserves a serious referee. I would send it to review with a clear request: either repair the Appendix A proof or explicitly delegate Theorem 2.5 to [18] and drop the flawed proof. The main new theorems are significant enough that the paper should be published after that fix.","headline":"Real new results and a useful counterexample, but the appendix proof of the geometric Littlewood-Offord theorem has a counting error that breaks the self-contained proof of the main multilinear theorem.","tokens_in":723,"tokens_out":742,"would_cite":true,"duration_ms":73868,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D40","60E15","15A69","11B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves optimal anticoncentration for multilinear polynomials (point probabilities $n^{-1+\\varepsilon}$ unless nearly reducible), power savings for complex quadratics, and disproves the original multilinear conjecture in degree…","keywords":["Littlewood–Offord problem","anticoncentration","Rademacher random variables","multilinear forms","partition rank","quadratic polynomials","inverse Littlewood–Offord","tensor rank property testing"],"falsifier":"For small $n$ (say up to 30), enumerate $3$-multilinear forms with coefficients in $\\{-1,0,1\\}$, retain only those whose every $4\\times 4\\times 4$ subtensor has partition rank at least 2, and compute the maximum value probability by exact enumeration over all sign configurations; exceeding $n^{-1+\\varepsilon}$ for any fixed $\\varepsilon>0$ would refute the flagship $n^{-1+\\varepsilon}$ theorem.","tokens_in":57832,"feed_emoji":"🎲","tokens_out":15758,"duration_ms":150060,"temperature":0.7,"pith_summary":"The paper studies how much a degree-$d$ polynomial of $n$ independent Rademacher variables can concentrate on a single value. It argues that the general $n^{-1/2}$-type bound can be dramatically improved unless the polynomial is algebraically structured, namely close to a reducible polynomial or to a low-rank quadratic form. As its flagship result, it shows that every $d$-multilinear form either is close to a product of lower-degree multilinear factors or has point probability at most $n^{-1+\\varepsilon}$, resolving a repaired form of the multilinear conjecture; since examples from the multiplication table problem show no bound below $(\\log n)^{\\alpha}/n$ is possible, the $n^{-1}$ exponent is optimal up to $n^{\\varepsilon}$. For complex quadratic polynomials that are robustly irreducible it proves a power saving with exponent $13/24$, and for general quadratics high rank of the quadratic part forces point probabilities near $n^{-1}$. An appendix disproves the original multilinear conjecture by exhibiting robustly irreducible degree-$d$ polynomials with point probability at least $\\varepsilon/n$.","feed_headline":"Multilinear forms concentrate at most 1/n unless they factor","feed_subtitle":"Only polynomials close to products can pile probability mass on one value; all others spread to n^{-1+\\varepsilon}.","key_machinery":"The machinery has five linked components. First, a local-to-global tensor lemma: if all but a $\\delta$-fraction of small $2^{d-1}\\times\\cdots\\times 2^{d-1}$ subtensors of a $d$-dimensional tensor are reducible, then changing an $\\varepsilon$-fraction of entries makes the whole tensor reducible, with $\\delta=(\\varepsilon/2)^{2^{d-1}}$; the symmetric-matrix analogue says many singular $r\\times r$ submatrices force near-low-rank while preserving symmetry. Second, a classification of the maximal linear subspaces of the variety of reducible tensors: each is a fixed tensor $T^\\star$ on one subset of the $d$ types, tensored with arbitrary tensors on the complementary types. Third, a geometric Littlewood–Offord theorem: a Rademacher sum of vectors lands in a proper variety with probability $\\ge n^{-1/2+\\varepsilon}$ only if almost all summands lie in a linear subspace whose translate is contained in the variety. Fourth, multi-copy decoupling inequalities that turn a quadratic anticoncentration event into $k$ simultaneous linear equations in independent random vectors, raising the exponent from $1/2$ toward $1$ as $k$ grows. Fifth, an optimal inverse linear Littlewood–Offord theorem, used to show the decoupled linear systems are unlikely to hit low-volume generalized arithmetic progressions.","core_discovery":"The paper's central claim is that algebraic structure, not coefficient sparsity, is the true obstruction to anticoncentration. For a $d$-multilinear form $f$ — a degree-$d$ polynomial whose variables split into $d$ types, one variable per type in each monomial — the coefficient tensor either can be made reducible (partition rank 1) by changing $\\varepsilon n^d$ entries, or else $\\sup_z \\mathbb{P}[f(\\xi_1,\\dots,\\xi_n)=z] \\le n^{-1+\\varepsilon}$; this is the repaired multilinear conjecture with the optimal $n^{-1}$ exponent. Over the complex numbers, a general quadratic polynomial that is not $\\varepsilon n^2$-close to any reducible quadratic satisfies the power saving $n^{-13/24+\\varepsilon}$, and if its quadratic part is not close to a form of rank less than $2k^2$, the bound improves to $n^{-1+2/k}$. The mechanism is local-to-global: sample small subtensors, classify the linear subspaces of the variety of reducible tensors, and reduce high-degree concentration to linear concentration via multi-copy decoupling and an inverse theorem for linear Littlewood–Offord. The original conjecture with $n^{-d/2}$ is false: the $d$-multilinear form $L_1\\cdots L_d - L_{d+1}\\cdots L_{2d}$ is robustly irreducible over $\\mathbb{C}$ yet has $\\mathbb{P}[f(\\xi)=0]\\ge \\varepsilon/n$, so no bound better than roughly $1/n$ is possible in general.","pith_inferences":["If the hidden constants in the imported inverse theorem were made explicit, the same induction could plausibly push the complex-quadratic exponent $13/24$ up toward $1$, matching the general conjecture that robust irreducibility alone yields $n^{-1+o(1)}$.","The counterexample family $P(L_1,\\dots,L_k)$ links the Littlewood–Offord problem to counting integral points on varieties: a transfer of dimension-growth estimates into this discrete setting would be a route toward structural inverse theorems, an implication the authors flag but do not develop.","The multi-copy decoupling inequalities look portable: applying the same $k$-copy trick to degree-$d$ polynomials, not just quadratics, should raise the achievable exponent toward $n^{-d/2}$ whenever the associated tensor has high enough rank or strength.","The local-to-global lemmas suggest practical constant-time rank tests: sampling $O(1)$ random subtensors decides whether a huge tensor is close to partition rank 1, and the symmetric-matrix analogue gives a symmetry-preserving low-rank approximation algorithm."],"forward_implications":["A $d$-multilinear form has point probability at most $n^{-1+\\varepsilon}$ unless it is $\\varepsilon n^d$-close to reducible, settling the multilinear case at the best possible exponent up to $n^{\\varepsilon}$.","Over $\\mathbb{C}$, every robustly irreducible quadratic polynomial satisfies $\\sup_z \\mathbb{P}[f(\\xi)=z] \\le n^{-13/24+\\varepsilon}$, a genuine power saving below the generic $n^{-1/2}$ bound.","For quadratic polynomials, guaranteeing that the quadratic part is not close to a form of rank less than $2k^2$ forces $\\sup_z \\mathbb{P}[f(\\xi)=z] \\le n^{-1+2/k}$; taking $k$ large approaches the $1/n$ barrier.","The example $f=L_1\\cdots L_d - L_{d+1}\\cdots L_{2d}$ is robustly irreducible even over $\\mathbb{C}$ yet assigns probability at least $\\varepsilon/n$ to $0$, so the original $n^{-d/2}$ conjecture fails for every $d\\ge 3$.","The local-to-global lemmas yield constant-query property tests: a tensor can be certified close to partition rank 1 by sampling constantly many small subtensors, and a symmetric matrix close to rank $r$ by sampling small principal submatrices."],"supporting_citations":[{"why":"Supplies the $d=2$ base case of the induction and the conjecture that the paper repairs.","marker":"[9]"},{"why":"The general $n^{-1/2+\\varepsilon}$ polynomial Littlewood–Offord bound that the structural results improve.","marker":"[40]"},{"why":"The imported optimal inverse Littlewood–Offord theorem used to control generalized arithmetic progressions in the quadratic proofs.","marker":"[45]"},{"why":"The high-dimensional linear Littlewood–Offord inequality used to estimate the probability of the decoupled linear events.","marker":"[23]"},{"why":"The geometric Littlewood–Offord theorem for varieties that underlies the variety version used in the multilinear induction.","marker":"[18]"},{"why":"The multi-copy decoupling scheme for quadratic Littlewood–Offord that motivates the decoupling inequalities used here.","marker":"[34]"},{"why":"Multiplication-table lower bounds showing the $n^{-1+\\varepsilon}$ exponent cannot be improved to $o(1/n)$.","marker":"[17]"}],"fun_headline_variants":["Algebraic structure, not sparsity, governs concentration","Multilinear forms concentrate only if close to products","Improved Littlewood-Offord bound ties to reducibility","Costello conjecture corrected: optimal 1/n bound","Polynomial concentration ruled by algebraic structure"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on an imported quantitative description of when a signed sum of vectors concentrates on one value, and the hidden constants in that description are not tracked; if those constants were substantially worse than stated, the exponents $13/24$ and $1-2/k$ would weaken.","fun_headline_variants_meta":{"raw":{"variants":["Algebraic structure, not sparsity, governs concentration","Multilinear forms concentrate only if close to products","Improved Littlewood-Offord bound ties to reducibility","Costello conjecture corrected: optimal 1/n bound","Polynomial concentration ruled by algebraic structure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000322,"raw_usage":{"total_tokens":1873,"prompt_tokens":1073,"completion_tokens":800,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":689,"completion_tokens_details":{"reasoning_tokens":726}},"tokens_in":689,"tokens_out":800,"duration_ms":8933,"temperature":1.0,"reasoning_tokens":726,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:49:13.601420+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For small $n$ (say up to 30), enumerate $3$-multilinear forms with coefficients in $\\{-1,0,1\\}$, retain only those whose every $4\\times 4\\times 4$ subtensor has partition rank at least 2, and compute the maximum value probability by exact enumeration over all sign configurations; exceeding $n^{-1+\\varepsilon}$ for any fixed $\\varepsilon>0$ would refute the flagship $n^{-1+\\varepsilon}$ theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the $d=2$ base case of the induction and the conjecture that the paper repairs."},{"cited_title":"12(2016), Paper No","cited_arxiv_id":null,"evidence_quote":"The general $n^{-1/2+\\varepsilon}$ polynomial Littlewood–Offord bound that the structural results improve."},{"cited_title":"Nguyen and V","cited_arxiv_id":null,"evidence_quote":"The imported optimal inverse Littlewood–Offord theorem used to control generalized arithmetic progressions in the quadratic proofs."},{"cited_title":"Halász,Estimates for the concentration function of combinatorial number theory and probability, Period","cited_arxiv_id":null,"evidence_quote":"The high-dimensional linear Littlewood–Offord inequality used to estimate the probability of the decoupled linear events."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The geometric Littlewood–Offord theorem for varieties that underlies the variety version used in the multilinear induction."},{"cited_title":"Ford,The distribution of integers with a divisor in a given interval, Ann","cited_arxiv_id":null,"evidence_quote":"Multiplication-table lower bounds showing the $n^{-1+\\varepsilon}$ exponent cannot be improved to $o(1/n)$."}],"review_version":1}