REVIEW 2 major objections 4 minor
Dense sets without large sumsets
T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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
What would settle it
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.
Extended reading notes
Core claim
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
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Theorem 3.1, Eq. (3.3)-(3.9)] 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 2, Lemma 2.1 and Eq. (2.8)] 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.
minor comments (4)
- [Eq. (2.9)] 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.
- [Eqs. (2.16)-(2.18)] 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).
- [Abstract and Theorem 1.1] 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 5, proof of Theorem 1.1] 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.
Circularity Check
No significant circularity: the random-set construction is derived from external fingerprint/GAP lemmas and standard additive combinatorics; the only self-citation is an independent structural theorem.
full rationale
Walking the derivation chain: Theorem 1.3 is the core probabilistic claim, and it is proved by partitioning pairs (A,B) into the three regimes |A+B| ≤ Ck, Ck < |A+B| < k(k+1)/2, and |A+B| ≥ k(k+1)/2, handled by Lemma 2.1, Corollary 3.3, and Lemma 4.1 respectively. The small-sumset estimate is not circular: Lemma 2.1 applies the external Theorem 1.4 ([3, Theorem 2]) to produce fingerprints A',B', and then (2.12) is derived from Theorem 1.4 together with Corollary 2.6 and the Plünnecke–Ruzsa inequality. No parameter is fitted to the desired conclusion: δ is an independent random density, and the threshold k=(3+γ)log n/log(1/δ) is a free theorem parameter. The only self-citation is Theorem 2.3, quoted from [4, Corollary 8.4] by Campos–Dahia–Marciano, two of whose authors overlap with the present paper. That theorem is genuinely load-bearing for the GAP-counting step, but it is a distinct published structural result with explicit hypotheses (2.2) that do not include the target statement, and it is not an ansatz or a uniqueness assertion; under the stated review rules it constitutes independent support rather than circularity. The possible hidden regime restriction in [3, Theorem 2] raised in the skeptical analysis is a correctness risk about an external theorem, not a circularity within this paper. The paper also explicitly notes a limitation in Section 6 (that these methods cannot resolve the limit question), but that is an optimality limitation, not a circular step. Overall, the claimed upper bound does not reduce to its inputs by construction, and no fitted input is renamed as a prediction.
Assumptions & free parameters
assumptions (10)
- domain assumption Theorem 1.4 = [3, Theorem 2] (Bollobás–Leader–Tiba): fingerprint subsets A′,B′ with |A′|,|B′|≤β√k and large |A′+B′|
- standard math Proposition 2.2 (Freiman lemma consequence): if |X+X|≤κ|X| then dim_F(X)≤2κ−1
- domain assumption Theorem 2.3 = [4, Corollary 8.4]: X with small doubling lies in a proper d-GAP of controlled size with d≤dim_F(X)
- standard math Theorem 2.4 (Plünnecke–Ruzsa inequality)
- standard math Theorem 2.5 (Ruzsa's asymmetric Freiman lemma) and Corollary 2.6
- standard math Lemma 3.2 (Green/Green–Morris counting of sets with bounded Freiman dimension)
- standard math Lemma 4.2 (Ruzsa covering lemma) and Corollary 4.3
- standard math Bertrand's postulate: prime p with 2n≤p≤4n
- standard math Chernoff bound for binomial concentration
- standard math Properness conversion for d-GAPs ([10, Theorem 2.1])
Cite this review
Pith. "Pith review of Dense sets without large sumsets." pith.science (2026). https://pith.science/paper/2H4A4YZQ
@misc{pith2026260715269,
author = {Pith},
title = {Pith review of: Dense sets without large sumsets},
year = {2026},
howpublished = {\url{https://pith.science/paper/2H4A4YZQ}},
note = {Machine review of arXiv:2607.15269}
}
abstract
We prove, for all fixed $0 < \delta < 1$, and all sufficiently large $n$, that there exists $S \subset [n]$ with $|S| \ge \delta n$ such that $A + B \not \subset S$ for all ${A, B \subset \mathbb{N}}$ satisfying $$\min\big\{|A|, |B|\big\} \ge \big(3 + o(1)\big) \frac{\log n }{ \log (1 / \delta)}.$$ A very recent result of Hern\'andez and Hetzel shows that our bound is sharp up to a factor of 3, and together our results settle a conjecture of Kra, Moreira, Richter, and Robertson. In fact, we prove that a $\delta$-dense random subset of $[n]$ is a valid choice for $S$ with high probability, and that one can take $n^{-\alpha} \le \delta \le 1 - c$ where $c > 0$ is fixed and $\alpha > 0$ depends only on the $o(1)$ error, answering another question of the same authors in a strong form.
Reviewed August 1, 2026 · model on record in the stance chip above.
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