{"id":"ecd526be-537c-48f0-be5c-c5b2846424ac","arxiv_id":"2411.14145","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Dense cube-avoiding summands in G^n are nearly contained in product sets over a bounded set of coordinates.","lead":"The paper proves that dense subsets of a high-dimensional finite abelian group whose sum avoids a fixed forbidden cube must be almost entirely encoded in a bounded number of coordinates. The result provides a structural dichotomy for cube-avoiding sumsets, with potential applications to additive combinatorics problems such as the Additive Basis Conjecture.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central proof imports Proposition 2.3/3.4 verbatim from [7, Thm 7.1]; if that theorem's pseudorandomness notion, uniform-marginal hypothesis, or multi-variable form differs from the statement used here, the structure theorem loses its only n-uniform ingredient.","rationale":"I read the main argument as sound conditional on the cited correlation theorem. The regularity lemma is a standard energy-increment argument, the epsilon-approximation counts are correct, and the rho(P)<1 checks in Sections 2.3 and 3.2 are coherent, including the reduction of the d-variable case to the two-variable case by conditioning. Thus I do not see an internal contradiction or a counterexample to the structural claim. The single serious risk is that the paper's statement of [7, Theorem 7.1] is not exactly the theorem in the literature. The reader identified the same concern, so my read does not change the CONDITIONAL verdict. I would not reject the paper, but I would insist on a line-by-line verification of the cited theorem before removing the conditionality. The secondary gap in Lemma 2.2 (forcing a non-empty I when the initial sets are already globally pseudorandom) is worth patching but is not the decisive issue, since in the main application Proposition 2.3 would already give the contradiction in that case.","tokens_in":12891,"tokens_out":23116,"duration_ms":225709,"concrete_test":"Compare the published text of [7, Theorem 7.1] (Discrete Analysis 19, 2018) with Theorems 2.5 and 3.5. Check: (i) are r, beta, c independent of n and uniform in the alphabet size, or must the alphabet be fixed before constants are chosen? (ii) does the pseudo-randomness condition agree with Definition 2.1 when P has uniform marginals, or is a different cylinder condition used? (iii) is the statement valid for an arbitrary number d of sources, or only d=2? If (iii) fails, attempt to derive Proposition 3.4 from the two-variable theorem by the conditioning argument in Section 3.2; if the derivation cannot be completed, Theorem 1.3 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every n-uniform step in the paper passes through Theorems 2.5 and 3.5, which are quoted as special cases of [7, Theorem 7.1]. Proposition 2.3 and Proposition 3.4 are direct corollaries of those theorems, and Theorems 1.1 through 1.3 rest on those propositions. The reductions around the black box (Lemma 2.2, the slice-density bookkeeping, and the verification that rho(P)<1) are internally coherent, so the load-bearing question is whether the import from [7] is faithful. Three specific points need checking. (1) [7] may define pseudo-randomness with respect to the marginal of P rather than the uniform density used in Definition 2.1; in the present applications the marginals of P are uniform, so this would be harmless, but the paper's theorem statement would then not be literally the cited theorem. (2) The constants r, beta, c in [7] may depend on the finite alphabet Omega or on a lower bound on n; the paper uses one theorem for all n and all finite abelian G, so any such dependence would break the uniform-in-n conclusion. (3) The d-variable form used for Theorem 3.5 is asserted without proof to be a special case of the same cited theorem; if [7, Thm 7.1] is only for two sources, the multi-summand extension has no external support. Because Proposition 3.4 is the only route from pseudorandom dense summands to a positive density of Z0^n-sums, failure at any of these points leaves Theorems 1.1-1.3 unproved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a structural theorem for dense subsets of G^n (G a finite abelian group) whose sumset avoids a power Z_0^n of a forbidden set Z_0. If Z_0 is not contained in any strict coset, then for any fixed number d >= 2 of summands and any epsilon > 0, any family E_1,...,E_d in G^n with empty (E_1+...+E_d) ∩ Z_0^n is epsilon-close to a family depending on a common set I of coordinates with 0 < |I| < C(d,G,epsilon), where C is independent of n. The proof combines a simultaneous regularity lemma (Lemma 2.2 and Lemma 3.3) with a pseudorandom-summands proposition (Proposition 2.3 and Proposition 3.4) that is deduced from an external correlation theorem of Hazła, Holenstein, and Mossel (Theorems 2.5 and 3.5). The paper also gives examples showing that the assumption on Z_0 is necessary, proves the two-summand case over Z_p and finite abelian groups, and concludes with open problems on optimal bounds and generalisations to latin squares.","tokens_in":13182,"tokens_out":11206,"duration_ms":103573,"significance":"If correct, this is a strong and clean structural result in high-dimensional additive combinatorics: the conclusion is uniform in n, the allowed coordinate set is common to all summands, and the non-degeneracy condition on Z_0 is shown to be necessary. The internal reductions are elegant and the energy bookkeeping in Lemma 2.2 is coherent. The paper is honest about its dependence on an external theorem, and the main risk is the faithfulness of that import rather than an internal error. The d-variable extension to several summands is a valuable contribution, but it rests entirely on the multi-variable form of the quoted correlation theorem.","major_comments":[{"comment":"The proof of Theorems 1.1–1.3 rests entirely on the import of [7, Theorem 7.1], stated as Theorems 2.5 and 3.5. The manuscript says these are 'special cases' but does not verify that the hypotheses match the source. In particular: (i) Definition 2.1 defines pseudorandomness with respect to the uniform density on X^n, whereas the theorem in [7] may define it with respect to the product of the marginals of the distribution P; in the present applications the marginals are uniform, so this is repairable, but the statement as written is not literally the cited theorem. (ii) The quantifier order in Theorem 2.5 asserts constants r, beta, c that are independent of the finite alphabet Omega and of n; if [7] allows dependence on Omega or on n, the applications still go through because G is fixed and the small-n case is trivial, but the quoted statement needs qualification to be faithful. (iii) Theorem 3.5 is asserted to be the d-variable special case of [7, Theorem 7.1], but no justification is given that the cited theorem covers d >= 3. Proposition 3.4, and hence Theorem 1.3, depends on this d-variable version. The authors should either provide a proof of Theorem 3.5 or give a precise statement of [7, Theorem 7.1] showing that it indeed implies the claimed multi-variable version. As it stands, this is a load-bearing gap in the proof of the main theorem.","section":"Section 2.3, Theorem 2.5; Section 3.2, Theorem 3.5"}],"minor_comments":[{"comment":"The proof uses the symbol p for the size of X without defining it; the energy-increment bound and the growth bound should be stated in terms of |X|. Also, the bound on |I_{s+1} \\ I_s| appears as 'p|I_s|r' in the typeset text; this should presumably be p^{|I_s|} r or |X|^{|I_s|} r.","section":"Lemma 2.2, proof"},{"comment":"The argument to ensure that I is non-empty is incomplete: if no slice is non-pseudorandom at the empty coordinate set, then the first iteration adds no coordinates and I remains empty. This is easily fixed by adding an arbitrary coordinate to I_0 at the start.","section":"Lemma 2.2, proof"},{"comment":"The symbol U_j is used both for a single coordinate and for the vector of all other coordinates in the definition of rho(P). This overloading is confusing and should be disambiguated.","section":"Section 3.2, definition of rho(P)"},{"comment":"The final sentence appears to be missing the conclusion '(E_1 + ... + E_d) ∩ Z_0^n = ∅'; the equality and the empty-set symbol are lost in the typesetting.","section":"Example 3.2"}],"recommendation":"major_revision","confidential_remarks":"The central concern is the accuracy of the quoted theorem [7, Theorem 7.1]. If the authors confirm that the theorem, in the exact form needed here, appears in the cited paper or provide a proof of the d-variable version, the paper is very likely acceptable. The two-summand part over Z_p and finite abelian groups is internally coherent and well-presented; the multi-summand theorem is the part that most needs verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you my read of Karam–Keevash, arXiv:2411.14145. The headline: this is a real theorem, the proof is coherent, and the only serious soft spot is the fidelity of the external correlation theorem they import.\n\nThe new content is a structural dichotomy: for d≥2, finite abelian G, and Z0 not contained in a strict coset, any dense E1,...,Ed in G^n whose sum avoids Z0^n are ε-close to sets depending on a common coordinate set I of size O(1) independent of n (Theorem 1.3). That is new. The proof is a two-step: a simultaneous regularity lemma from [9] finds I such that most slices are pseudorandom; then the correlation theorem of Hazła–Holenstein–Mossel [7] shows that pseudorandom dense slices cannot avoid Z0^I. The density bookkeeping and the verification that ρ(P)<1 are careful, and Examples 3.1–3.2 show the coset condition is necessary.\n\nThe soft spots. The main one: the proof rests word-for-word on [7, Thm 7.1], stated as Theorems 2.5 and 3.5. The authors call these 'special cases' but never check the hypotheses: is pseudorandomness in [7] with respect to the uniform measure or the marginal of P? In the applications the marginals are uniform, so likely harmless. Do r,β,c depend on n or the alphabet? The paper needs a uniform-in-n statement. And Theorem 3.5, the d-variable version, is asserted to be a special case; if [7] only proves two sources, the multi-summand extension lacks support. This is a genuine referee issue, but not a rejection: the original theorem probably covers these cases, and the burden is on the authors to state the exact version and verify compatibility.\n\nThere is a minor gap in Lemma 2.2 when both sets are globally pseudorandom: the iteration can get stuck at I=∅. In the main application this cannot happen because then the correlation theorem would immediately contradict the avoidance assumption, but the lemma needs a small fix. A few typos (e.g., 'p|Is|r' should be 'p^{|Is|}r') are easy to clean up.\n\nWho should read this? Anyone working on dense subsets of product spaces or on regularity-plus-correlation arguments. It is not a breakthrough, but it is a solid, usable theorem. I would send it to a serious referee, asking them to nail down the [7] import and patch Lemma 2.2. Conditional acceptance is the right verdict.","headline":"A genuinely new bounded-dimensional dichotomy for cube-avoiding sumsets, with a sound proof that hinges on an external correlation theorem the authors need to state more carefully.","tokens_in":13780,"tokens_out":8765,"would_cite":true,"duration_ms":81396,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B13","05D05","05D40","20K01"],"pacs":[],"model":"deepseek-v4-flash","headline":"Dense summands avoiding a cube in a finite abelian group power are almost entirely captured by a bounded common coordinate set.","keywords":["sumsets","cube avoidance","finite abelian groups","pseudorandomness","regularity lemma","additive combinatorics","high-dimensional structure","correlation"],"falsifier":"Construct, for some finite abelian $G$ and some $Z_0$ not contained in a strict coset, a sequence of dense $E,F\\subset G^n$ with $(E+F)\\cap Z_0^n=\\emptyset$ but with no nonempty $I\\subset[n]$ of size below the claimed $C(G,\\varepsilon)$ such that both sets are $\\varepsilon$-close to cylinders over $I$; Proposition 2.3 says such examples cannot exist, so the first such example would refute the theorem. A cheaper falsifier is to check the imported correlation theorem directly: exhibit dense pseudorandom $f,g$ on i.i.d. coordinates with $\\rho(P)<1$ and $\\mathbb{E}fg$ arbitrarily small, which would contradict Theorem 2.5.","tokens_in":12621,"feed_emoji":"🧊","tokens_out":8685,"duration_ms":79227,"temperature":0.7,"pith_summary":"This paper proves a structural rigidity statement for dense subsets of a large finite abelian group power. If $Z_0\\subset G$ is not contained in any proper coset and dense sets $E_1,\\dots,E_d\\subset G^n$ have a sumset that avoids the cube $Z_0^n$, then all of the $E_j$ are almost entirely contained in cylinders $E'_j\\times G^{I^c}$ over one common coordinate set $I$, whose size is bounded independently of $n$. Up to error $\\varepsilon |G^n|$, the whole high-dimensional configuration is explained by a low-dimensional configuration on $I$ that again avoids $Z_0^I$. This matters because it turns an avoidance constraint in high dimension into a finite classification problem, and it targets phenomena such as additive bases of vector spaces where cube-avoiding sums appear.","feed_headline":"Dense sums that dodge a cube must be nearly low-dimensional","feed_subtitle":"For finite abelian groups, a forbidden cube in the sumset forces one bounded coordinate set to explain almost everything.","key_machinery":"Two tools carry the argument. The first is a simultaneous regularity lemma (Lemma 2.2): given accuracy parameters, any finite family of subsets of $G^n$ has a nonempty coordinate set $I$ of size bounded in terms of the parameters alone such that most coordinate slices of every set are $(r,\\beta)$-pseudorandom, meaning restrictions to at most $r$ coordinates change density by at most $\\beta$. The second is a correlation theorem for product-space models (restated as Theorems 2.5 and 3.5): dense pseudorandom Boolean functions on independent identically distributed coordinates whose joint distribution has correlation strictly below $1$ must have positive expected product. Applied to the distribution supported on tuples with sum in $Z_0$, this yields the key proposition that dense pseudorandom summands contain a positive fraction of tuples whose sum lands in $Z_0^n$. The contradiction argument then shows the dense pseudorandom slices over $I$ cannot have sum in $Z_0^I$, forcing the cylinders to avoid it.","core_discovery":"The paper's central claim is Theorem 1.3: for $d\\ge 2$, every finite abelian group $G$, every $Z_0\\subseteq G$ not contained in a strict coset, and every $\\varepsilon>0$, there is a constant $C=C(d,G,\\varepsilon)$ such that whenever $E_1,\\dots,E_d\\subseteq G^n$ satisfy $(E_1+\\cdots+E_d)\\cap Z_0^n=\\emptyset$, there is a nonempty $I\\subseteq[n]$ with $|I|<C$ and subsets $E'_j\\subseteq G^I$ such that $|E_j\\setminus(E'_j\\times G^{I^c})|\\le \\varepsilon|G^n|$ for all $j$ and $(E'_1+\\cdots+E'_d)\\cap Z_0^I=\\emptyset$. In words, dense summands that jointly avoid the cube are $\\varepsilon$-close to sets depending only on a common bounded family of coordinates; the theorem's content is that this bounded family can be chosen before seeing $n$.","pith_inferences":["Beyond the paper: the same regularity-plus-correlation scheme should apply to any tensor-product constraint $f^{\\otimes n}(E_1,\\dots,E_d)\\cap Z_0^n=\\emptyset$ for which the uniform distribution on the solution set of $f(x_1,\\dots,x_d)\\in Z_0$ has correlation $<1$; the paper raises Latin squares as a candidate, and the bottleneck would be verifying the correlation condition.","Beyond the paper: if the correlation theorem can be made effective with polynomial dependence, the same proof would give a bound on $|I|$ polynomial in $\\log(1/\\varepsilon)$, bearing on the paper's Question 4.1, which asks specifically for $|I|=O(\\log(\\varepsilon^{-1}))$.","Beyond the paper: the theorem implies a form of finitary stability for cube-avoiding dense configurations at every fixed accuracy; combining it with a classification of the low-dimensional avoiders in $G^I$ would yield a complete approximate description of all dense avoiders in arbitrary dimension."],"forward_implications":["For any fixed $G$ and $Z_0$, the family of dense $d$-tuples avoiding $Z_0^n$ is, up to $\\varepsilon$-error, parameterised by finitely many coordinate sets $I$ and low-dimensional avoiders in $G^I$; the classification is independent of $n$.","When one summand is sparse the theorem degenerates to a trivial bound, and otherwise all summands must share the same structured coordinate set; in particular no dense example can split structure across disjoint coordinate sets while avoiding $Z_0^n$.","The condition on $Z_0$ is necessary: if $Z_0$ lies in a proper coset of a subgroup, Examples 3.1 and 3.2 produce high-dimensional avoiders with no bounded common coordinate structure.","For the prime two-summand case, if $E+F$ avoids $\\{0,1\\}^n$ then $E$ and $F$ are $\\varepsilon$-close to $E'\\times\\mathbb{Z}_p^{I^c}$ and $F'\\times\\mathbb{Z}_p^{I^c}$ for a common nonempty $I$ of size bounded by $C(p,\\varepsilon)$."],"supporting_citations":[{"why":"Supplies the product-space correlation theorem restated as Theorems 2.5 and 3.5; the entire reduction to pseudorandom summands depends on this external result.","marker":"[7]"},{"why":"Supplies the simultaneous regularity lemma pattern used in Lemma 2.2 to find the bounded coordinate set $I$ with mostly pseudorandom slices.","marker":"[9]"},{"why":"States the Additive Basis Conjecture that makes cube-avoiding sumsets in $\\mathbb{Z}_p^n$ a motivated target; the bounded-dimension theorem is designed as structural input for this problem.","marker":"[1]"}],"fun_headline_variants":["Cube avoiders: dense sumsets live near a bounded coordinate slice","Dense cube-avoiding sums are forced into few coordinates","Sums that miss a cube must be compressible to fixed coordinates","For every dense cube-free sum, a bounded core coordinates emerge","Cube-free sums: dense sets are near a common fixed subspace"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof imports a theorem saying that dense pseudorandom Boolean functions on independent coordinates with correlation below one must have positive expected product; the paper restates this result rather than proving it, and if the restatement misses a hypothesis (such as a restriction on the dimension relative to the pseudorandomness parameters) the structure theorem would lose its main support.","fun_headline_variants_meta":{"raw":{"variants":["Cube avoiders: dense sumsets live near a bounded coordinate slice","Dense cube-avoiding sums are forced into few coordinates","Sums that miss a cube must be compressible to fixed coordinates","For every dense cube-free sum, a bounded core coordinates emerge","Cube-free sums: dense sets are near a common fixed subspace"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000348,"raw_usage":{"total_tokens":1903,"prompt_tokens":946,"completion_tokens":957,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":869}},"tokens_in":562,"tokens_out":957,"duration_ms":9216,"temperature":1.0,"reasoning_tokens":869,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:30:44.857646+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct, for some finite abelian $G$ and some $Z_0$ not contained in a strict coset, a sequence of dense $E,F\\subset G^n$ with $(E+F)\\cap Z_0^n=\\emptyset$ but with no nonempty $I\\subset[n]$ of size below the claimed $C(G,\\varepsilon)$ such that both sets are $\\varepsilon$-close to cylinders over $I$; Proposition 2.3 says such examples cannot exist, so the first such example would refute the theorem. A cheaper falsifier is to check the imported correlation theorem directly: exhibit dense pseudorandom $f,g$ on i.i.d. coordinates with $\\rho(P)<1$ and $\\mathbb{E}fg$ arbitrarily small, which would contradict Theorem 2.5.","supporting_citations":[{"cited_title":"Haz ˛ła, T","cited_arxiv_id":null,"evidence_quote":"Supplies the product-space correlation theorem restated as Theorems 2.5 and 3.5; the entire reduction to pseudorandom summands depends on this external result."},{"cited_title":"Keevash, N","cited_arxiv_id":null,"evidence_quote":"Supplies the simultaneous regularity lemma pattern used in Lemma 2.2 to find the bounded coordinate set $I$ with mostly pseudorandom slices."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the Additive Basis Conjecture that makes cube-avoiding sumsets in $\\mathbb{Z}_p^n$ a motivated target; the bounded-dimension theorem is designed as structural input for this problem."}],"review_version":1}