{"id":"56b443f8-2ce5-4872-a492-c4f5aaba266e","arxiv_id":"2607.15269","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A δ-dense random subset of [n] almost surely contains no sumset A+B unless one side has size below about 3 log n/log(1/δ), matching the known lower bound up to factor 3.","lead":"This paper proves a conjecture in additive combinatorics: for any fixed density δ, one can—and a random set does—choose a δ-dense subset of {1,…,n} that contains no large sumset A+B. It pins the size threshold to within a factor of 3 and leaves one open question about the exact constant.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The small-sumset case (Lemma 2.1) depends wholly on the quoted [3, Theorem 2]; if that theorem has a hidden size or group restriction, the fingerprint step and Theorem 1.3 collapse.","rationale":"The reader accepted with moderate confidence and flagged the same external theorems. I agree that the most fragile point is Theorem 1.4; the entire small-sumset branch of the proof is a black box. Unlike [4, Corollary 8.4], whose hypotheses are checked explicitly in the proof, the application of [3, Theorem 2] is stated as a universal theorem and no verification of its regime is provided. I did not find an internal contradiction, and if the quotation is accurate the proof works. However, because the main novelty rests on this preprint result, acceptance should be conditional on confirming its exact hypotheses in Z/pZ, especially for constant k at δ=n^{-α}. This is a verification step, not a demonstrated flaw.","tokens_in":14475,"tokens_out":40391,"duration_ms":300022,"concrete_test":"Open arXiv:2206.09366 and check Theorem 2 verbatim against Theorem 1.4. Verify (a) β depends only on σ,ε and not on k; (b) the statement is for arbitrary finite abelian groups, including Z/pZ; (c) there is no lower bound on k or on |A+B|. Then rerun the proof of Lemma 2.1 in the boundary case δ=n^{-α} (so k=(2+γ)/α fixed) to confirm the fingerprint sets exist. If any of (a)–(c) fails, Theorem 1.3 is not proved; if they pass, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central proof's load-bearing step is Lemma 2.1. For pairs A,B⊂Z/pZ with |A+B|≤Ck it invokes Theorem 1.4 ([3, Theorem 2]) to produce fingerprints A′,B′ with |A′|,|B′|≤β√k and |A′+B′|≥(1−ε)|A*+B*| (equation 2.8). This is the only place the small-sumset range is controlled; the subsequent GAP/Chang counting cannot start without it. The application requires the theorem to hold uniformly for all k-subsets of Z/pZ, and at the lower end δ=n^{-α}, k=(2+γ)/α is a constant depending only on the chosen α. If [3, Theorem 2] actually requires k≥k0(σ,ε), or is proved only for torsion-free groups rather than Z/pZ, or has a large-sumset hypothesis, then (2.8) is unjustified exactly in the regime needed for Corollary 1.2. I found no internal error in the rest of the proof; the arithmetic and GAP-counting steps are coherent if Theorem 1.4 holds as quoted.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a quantitative finite analogue of the sumset conjecture of Kra, Moreira, Richter, and Robertson. The main result, Theorem 1.1, asserts that for every 0<γ≤1 and c>0 there is α>0 such that for all sufficiently large n and n^{-α}<δ≤1-c, there exists S⊆[n] with |S|≥δn and with no A+B⊆S whenever min{|A|,|B|}≥(3+γ)log n/log(1/δ). The stronger Theorem 1.3 says that a δ-random S has this property with probability tending to 1. Combined with the lower bound of Hernández and Hetzel, this determines φ(δ,n)/log n up to a factor 3+o(1), settling Conjecture 4.10 and answering Question 4.12 of Kra--Moreira--Richter--Robertson. The proof splits pairs (A,B) by |A+B|: small sumsets are handled in Section 2 via a fingerprint argument using the Bollobás--Leader--Tiba theorem, Chang-type GAP counting, and a Ruzsa covering argument; moderate sumsets are counted in Section 3; very large sumsets are handled in Section 4 by a simple union bound from Ruzsa's covering lemma.","tokens_in":14808,"tokens_out":22392,"duration_ms":163282,"significance":"If the proof is correct, this is a substantial contribution to quantitative additive combinatorics. It resolves a conjecture that was explicitly left open in the literature and identifies the correct growth rate up to the universal factor 3. The random-set formulation is strong and natural, and the proof strategy extends the Green--Morris random Cayley sum graph toolbox to the asymmetric setting. The paper is honest about the limits of the method: it cannot achieve the conjectured 2+o(1) constant because the Green--Morris large-sumset techniques do not transfer to A≠B. The main elements of the proof are explicitly stated and the arithmetic estimates in Sections 2--5 are internally coherent, provided the quoted external theorems hold in the required regimes.","major_comments":[{"comment":"The second bound (3.4) is stated for every k≥2200 when m≤k^{1+ξ}. However, the proof of (3.9) needs the additional condition k^{1/30-2ξ}≥6 in order to apply Lemma 3.2's second estimate with s=3m^2/k. This is not a consequence of k≥2200: with ξ=2^{-8}, 2200^{1/30-2ξ} ≈ 1.22, far below 6. Thus Theorem 3.1 as stated is not established. The later application in Corollary 3.3 can be repaired, since α is eventually chosen small enough to force k^ξ≥4/γ, which amply implies the missing condition, but the theorem statement and proof should be made consistent.","section":"Theorem 3.1, Eq. (3.3)-(3.9)"},{"comment":"The small-sumset case, and therefore the whole proof of Theorem 1.3, depends entirely on the quoted Bollobás--Leader--Tiba theorem (Theorem 1.4) to obtain fingerprints A',B' with |A'+B'|≥(1-ε)|A*+B*|. The application is in Z/pZ, with k only bounded below by a constant depending on α, and no other hypothesis is stated. Since the theorem is quoted without proof, I could not independently verify that it has no hidden lower bound on |A|,|B|, no large-sumset hypothesis, and no restriction to torsion-free groups. If any such restriction is present, (2.8) fails exactly where it is load-bearing. I ask the authors to state the precise hypotheses of [3, Theorem 2] and confirm explicitly that they hold in the regime used here.","section":"Section 2, Lemma 2.1 and Eq. (2.8)"}],"minor_comments":[{"comment":"The bound |A+A|≤C^2k is usually derived from Ruzsa's triangle inequality, not directly from the Plünnecke--Ruzsa inequality as Theorem 2.4 is labeled. The citation or the statement should be adjusted to avoid a mismatch.","section":"Eq. (2.9)"},{"comment":"The passage from binomial coefficients inom{|P|}{β√k} to the exponential form suppresses factors of e and β√k. These are indeed absorbed by 'adequately increasing C''', but the intermediate inequality should be spelled out, especially since the exponent (2.18) involves only 2β√k log(3C''k).","section":"Eqs. (2.16)-(2.18)"},{"comment":"The abstract says 'for all fixed 0<δ<1', while the theorem requires n^{-α}<δ≤1-c. The abstract should match the precise statement, or the earlier phrasing should be qualified.","section":"Abstract and Theorem 1.1"},{"comment":"The existence of δ' with δ<δ'≤1-c/2 and the inequality (3+γ')/log(1/δ')≤(3+γ)/log(1/δ) is asserted without proof. It follows by taking δ' sufficiently close to δ from above, but a short sentence making this explicit would help the reader.","section":"Section 5, proof of Theorem 1.1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript's central argument is in good shape if the quoted external results are sound. The main internal issue is the overbroad statement of Theorem 3.1, which is fixable locally. Given that Lemma 2.1 rests wholly on [3, Theorem 2], I recommend that the editor have the handling editor or a second referee verify that theorem's exact hypotheses and applicability in Z/pZ. I did not find a circularity or a hidden dependence on the target conjecture."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper closes a named conjecture: it shows that δ-dense subsets of [n] avoid k-size sumsets for k = (3+o(1)) log n / log(1/δ), matching Hernández–Hetzel's lower bound up to a factor of 3. That alone makes it a significant addition to the quantitative additive combinatorics program. The proof is thorough, and the random-set version (Theorem 1.3) is a genuinely stronger statement, not just a technical add-on.\n\nWhat's new: Theorem 1.1/1.3 give the first upper bound on the critical growth rate, and the random-set approach is the natural one. The argument splits into three sumset-size regimes, using Green's Freiman-isomorphism counting for moderate sizes and Ruzsa covering for large ones. The most delicate part is the small-sumset range (Lemma 2.1), where the paper invokes [3, Theorem 2] to produce 'fingerprints' A', B' with size O(√k) and sumset nearly as large as A*+B*. That invocation is load-bearing. The stress-test worry is that [3, Theorem 2] might carry a hidden regime restriction or group-theoretic constraint. But the quoted statement is fully general: finite non-empty subsets of an Abelian group. If that quote is accurate, the application to k-subsets of Z/pZ is legitimate, and the rest of the counting works. I see no internal contradiction or fitted parameter.\n\nThe soft spots are the reliance on two external black boxes ([3, Theorem 2] and the authors' own [4, Corollary 8.4]), and the unavoidable factor-3 gap. The paper is honest about the latter, even noting that random sets cannot beat factor 2. The dependence on α makes n astronomically large in extreme regimes, but that is permitted by the statements.\n\nOverall this is a strong paper. The proof is coherent, the claim is new, and the citation pattern is clean. The main risk is whether [3, Theorem 2] is exactly as quoted; if so, the theorem is settled. I'd send it to a serious referee without hesitation.","headline":"Settles Conjecture 4.10 with a (3+o(1)) upper bound and a random-set proof; the only real fragility is the black-box invocation of [3, Theorem 2], which as quoted has no hidden restriction.","tokens_in":15342,"tokens_out":5282,"would_cite":true,"duration_ms":36921,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D40","11B13","11P70"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every dense subset of [n], there is a dense S containing no sumset A+B with min{|A|,|B|} at least (3+o(1)) log n / log(1/δ), and a random dense S works with high probability.","keywords":["dense sets","sumsets","additive combinatorics","random subsets","probabilistic method","additive dimension","generalized arithmetic progressions","sumset avoidance"],"falsifier":"Simulate δ-random sets S⊂[n] for n=10^6 and δ=1/2, searching for A,B with min{|A|,|B|} ≥ (3+γ) log n / log 2 and A+B⊂S; Theorem 1.3 says such pairs occur with probability o(1). A routine appearance of such pairs would disprove the main claim. Alternatively, test the fingerprint theorem directly: for random k-sets A,B modulo a prime with |A+B|≤Ck, look for small subsets A',B' of size O(√k) with |A'+B'| ≥ (1-ε)|A*+B*|; their absence would invalidate Lemma 2.1.","tokens_in":14375,"feed_emoji":"🎲","tokens_out":11800,"duration_ms":87967,"temperature":0.7,"pith_summary":"This paper proves that dense subsets of {1,...,n} can be built — and in fact a random dense subset almost surely is one — that contain no sumset A+B of two sets whose sizes are at least (3+o(1)) log n / log(1/δ). Together with a recent lower bound showing every δ-dense set must contain sumsets of size (1-o(1)) log n/log(1/δ), this pins the correct logarithmic growth-rate threshold to a factor of 3 and settles a conjecture from a 2025 problem list on quantitative sumset configurations. The proof splits all pairs (A,B) by the size of their sumset and uses a fingerprint/arithmetic-progression counting argument, extending earlier techniques from the symmetric A=A case to asymmetric A≠B pairs.","feed_headline":"Random dense sets dodge every large sumset","feed_subtitle":"The logarithmic threshold is pinned to within a factor of three of optimal.","key_machinery":"The proof partitions all pairs A,B⊂[n] with |A|=|B|=k according to the size of |A+B|. In the small-sumset regime it invokes a quoted structural theorem to find 'fingerprints': tiny subsets A'⊂A, B'⊂B of size O(√k) whose sumset A'+B' is still at least (1-ε)|A*+B*|, where A*,B* are almost all of A,B. An asymmetric version of the classical structural lemma for sumsets then forces the additive dimension of A∪B — the largest lattice dimension into which the set embeds while preserving additive relations — to be bounded by a constant depending only on the sumset-size constant; a quantitative structure theorem places A∪B inside a small proper generalized arithmetic progression (GAP, a structured se","core_discovery":"At its core, the paper establishes Theorem 1.3: for every fixed γ>0 and c>0, if S⊂[n] is obtained by keeping each element independently with probability δ (with n^{-α}<δ≤1-c for a suitable α), then the probability that S contains a sumset A+B with min{|A|,|B|} ≥ (3+γ) log n / log(1/δ) tends to 0 as n grows. Since such a random S has size at least δn with high probability, this yields Theorem 1.1: a dense set with the same avoidance property exists. The defining quantity φ(δ,n) — the largest integer such that every δ-dense subset of [n] contains a sumset of that size — therefore satisfies (1-γ) log n / log(1/δ) ≤ φ(δ,n) ≤ (3+γ) log n / log(1/δ) for all large n, which settles the conjecture an","pith_inferences":["The factor-3 gap between the new upper bound and the known lower bound is unlikely to be closed by random sets: the paper itself observes that the random construction caps at constant 2, so a better upper bound would need a different, non-random construction.","The fingerprint technique may generalize to other settings where a dense set must avoid a structured configuration of the form X+Y, for instance in groups other than the integers or in higher dimensions.","One could probe the sharpness of the constants computationally: for fixed small n and δ, search for the largest sumset that fits inside a random S; the theorem predicts the threshold (3±o(1)) log n/log(1/δ), and observed deviations would signal a hidden constant issue.","The proof's dependence on a quoted structural theorem for fingerprints is a possible boundary: if that theorem's constants degrade badly with the group or with ε, the small-sumset estimate would need a different argument."],"forward_implications":["The theorem settles the 2025 conjecture: every δ-dense subset of [n] contains a sumset of size at least (1-γ) log n/log(1/δ), and some δ-dense sets contain no sumset above (3+γ) log n/log(1/δ).","A δ-random subset of [n] is, with high probability, an extremal example, so 'the typical dense set' avoids large sumsets; explicit constructions are not needed.","The asymmetric counting method for A+B (rather than A+A) is new and provides a tool for future sumset-containment problems in additive combinatorics.","In the symmetric case A=B, δ=1/2, the method reproduces the known optimal constant 2 for the largest set A with A+A⊂S in a random S; the paper notes in Section 6 that this is why the random-set approach cannot improve the upper-bound constant below 2."],"fun_headline_variants":["Dense sets can dodge all large sumsets","Random dense sets sidestep big sumsets","Sumset avoidance: density beats structure","Dense sets with no large sumsets exist"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that a quoted external theorem supplies, for every pair A,B with small sumset modulo a large prime, fingerprint subsets of size O(√k) whose sumset has size at least (1-ε)|A*+B*|, and that the authors' own earlier GAP-containment theorem covers the entire stated density range.","fun_headline_variants_meta":{"raw":{"variants":["Dense sets can dodge all large sumsets","Random dense sets sidestep big sumsets","Sumset avoidance: density beats structure","Dense sets with no large sumsets exist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000165,"raw_usage":{"total_tokens":1111,"prompt_tokens":795,"completion_tokens":316,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":258}},"tokens_in":539,"tokens_out":316,"duration_ms":3052,"temperature":1.0,"reasoning_tokens":258,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T23:42:21.250871+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate δ-random sets S⊂[n] for n=10^6 and δ=1/2, searching for A,B with min{|A|,|B|} ≥ (3+γ) log n / log 2 and A+B⊂S; Theorem 1.3 says such pairs occur with probability o(1). A routine appearance of such pairs would disprove the main claim. Alternatively, test the fingerprint theorem directly: for random k-sets A,B modulo a prime with |A+B|≤Ck, look for small subsets A',B' of size O(√k) with |A'+B'| ≥ (1-ε)|A*+B*|; their absence would invalidate Lemma 2.1.","supporting_citations":[],"review_version":1}