Additive structure in convex sets

Jakob F\"uhrer; Oliver Roche-Newton; Thomas F. Bloom

REVIEW 2 major objections 4 minor 17 references

Additive structure in convex sets

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper constructs arbitrarily large convex sets—finite real sets with strictly increasing consecutive gaps—that nonetheless contain Ω(|A|^{3/2}) non-trivial three-term arithmetic progressions, the largest count compatible with the standa

desk verdict The headline construction is new and probably correct, but the printed proof has a few fixable gaps and a wrong contrapositive in the introduction. read the letter →

arxiv 2509.01568 v1 pith:2VXPKDEP submitted 2025-09-01 math.CO math.NT

Thomas F. Bloom , Jakob Führer , Oliver Roche-Newton This is my paper · ORCID

classification math.COmath.NT MSC 11B3011B2505D40

keywords convexsetsthree-termarithmeticprogressionsadditiveenergycombinatoricsSidonsumsetrepresentationslattice-pointconstruction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Convex sets—finite sets of real numbers whose consecutive gaps grow—are usually expected to be additively unstructured, so it is natural to ask how many three-term arithmetic progressions they can contain. This paper proves the count can be large: it constructs convex sets A of arbitrarily large size with Ω(|A|^{3/2}) non-trivial 3-term arithmetic progressions. That exponent is the ceiling imposed by the standard Cauchy–Schwarz route from the additive-energy conjecture, so together with the unconditional upper bound T3(A) ≪ |A|^{5/3} proved here, the possible range for the exponent is narrowed to between 3/2 and 5/3. The paper also reproves the known bound that every element of A+A has O(|A|^{2/3}) representations, and constructs convex sets showing this representation bound is sharp. It further proves every convex set contains a Sidon subset of size |A|^{77/150−o(1)} and constructs convex sets where the largest Sidon subset has size at most |A|^{3/4}.

What carries the argument

The load-bearing object is the blocked quadratic family f_ℓ(x) = a x^2 + ℓ(x+n). The product term ℓ(x+n) is bilinear, so a fixed-k slice {f_ℓ(k)} is an arithmetic progression in ℓ; the tiny quadratic term a x^2 is added only to make each block convex and to control the gaps between blocks. Convexity of the union reduces to three displayed gap inequalities at each block boundary, and the residue choice n + y_ℓ ≡ 1 (mod ℓ) makes them collapse algebraically to conditions that a sufficiently small a satisfies. For the Sidon upper bound, the machinery is a known sumset construction in which B+C contains a convex set of quadratic size; a large Sidon subset would force a 4-cycle in a bipartite grap

What would settle it

Fix n = 8m^2, choose y_ℓ in [1,ℓ] satisfying n+y_ℓ ≡ 1 (mod ℓ), and solve the finite system of gap inequalities for a > 0. If for any m the system has no solution, the proof of Theorem 6 fails at its convexity step; if a solution exists for all tested m but the resulting sets have T3(A)/|A|^{3/2} → 0, the claimed Ω bound is not realized by this construction.

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Extended reading notes

Core claim

The central claim is Theorem 6: for any n,m with n ≥ 8m^2 there is a convex set A of size Θ(n) containing at least n/4m disjoint arithmetic progressions of length m. The set A is a union of blocks B_ℓ={f_ℓ(k): y_{ℓ-1}≤k≤x_ℓ} with f_ℓ(x)=a x^2+ℓ(x+n) and a>0 tiny. Each block is convex, and the residue choice n+y_ℓ≡1 (mod ℓ) makes the inter-block gap inequalities hold, so the union is convex. For fixed k, the f_ℓ(k) form an arithmetic progression; with n=Θ(m^2) this yields ≫|A|^{3/2} non-trivial three-term progressions. A related construction gives a convex set with an element centred in Ω(|A|^{2/3}) three-term progressions, so the known representation bound r_{A+A}(x)≪|A|^{2/3} is optimal.

Load-bearing premise

The whole construction hinges on choosing one tiny positive number a so that every block is convex and every gap between neighbouring blocks is ordered correctly at the same time; the proof does not pin down the left endpoint of the first block, and if such a uniform choice of a does not exist for some n,m the union need not be convex.

Editorial extensions

If this is right

The upper-bound problem for T3(A) on convex sets is narrowed to exponents between 3/2 and 5/3; the new construction shows the lower end is achievable.
The known representation bound r_{A+A}(x) ≪ |A|^{2/3} is optimal, so any sharper T3 upper bound cannot come from a better uniform representation bound.
For every fixed k ≥ 3, convex sets exist with ≫ |A|^{3/2} non-trivial k-term arithmetic progressions; the same construction applies to any fixed translation-invariant homogeneous linear equation.
The probabilistic proof yields a Sidon subset of size |A|^{77/150−o(1)} in every convex set, and would give |A|^{2/3−o(1)} if the additive-energy conjecture holds.
There are convex sets in which every subset of size ≫ |A|^{3/4} contains a non-trivial additive quadruple, so the largest Sidon subset can be no larger than |A|^{3/4} in those sets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The construction by itself does not refute the additive-energy conjecture: from T3(A) ≥ c|A|^{3/2} and the Cauchy–Schwarz inequality T3(A) ≤ |A|^{1/2}E(A)^{1/2} one only obtains E(A) ≥ c^2|A|^2, which is compatible with the conjectured |A|^{2+o(1)} energy; the conjecture's status remains open.
The feasibility of the gap inequalities is a finite linear program in the parameter a; testing n = 8m^2 for increasing m and checking whether any positive a survives would quickly show whether the unwritten left endpoint of the first block is a harmless omission or a real gap in the proof.
The union-of-blocks idea is not specific to quadratics: any family of convex functions whose values at each fixed k form an arithmetic progression, with gaps controllable by a parameter, would yield the same T3 lower bound; higher-degree convex choices may shift the trade-off between block length and block count and could expose where the 3/2 ceiling really comes from.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main result is real: a convex set with T3(A) = Omega(|A|^{3/2}), matching the Cauchy-Schwarz heuristic up to a constant. The construction via f_l(x) = a x^2 + l(x+n) is genuinely new and worth knowing. The paper also gives an alternative proof of Schoen's r_{A+A} bound using Erdos-Szekeres, shows that bound is tight, and proves new Sidon-set results. The Jarnik connection in Section 4 is a nice touch and places the construction in a broader context.

The theorems are plausibly correct, but the printed proof needs attention. In Theorem 6, B_m uses y_{m-1}, which is never defined; that's a typo but a reader cannot verify the first block as written. More substantively, convexity of the union A is asserted to follow from inequalities that hold for "sufficiently small a", but no uniform bound on a is produced. The stress-test note computes that a ~ m^2/n^2 should work for all l simultaneously; I checked the worst-case l=m and agree. So the gap is likely repairable, but as printed it is a real gap, not just a skipped routine detail.

Theorem 15 has an empty interval (x_{i+1}, x_i) where it should read (x_i, x_{i+1}); clearly a typo. And the introduction's "contrapositive" of Conjecture 1 is wrong: T3(A) >= |A|^{3/2} only gives E(A) >> |A|^2, which does not refute E(A) <= |A|^{2+o(1)}. The paper later states the correct belief that T3(A) << |A|^{3/2+o(1)} is the plausible upper bound, so the bad sentence should simply be deleted or rephrased.

None of these issues are load-bearing in the sense of making the main theorem false. The construction is explicit, the cited tools are standard, and the arguments are otherwise self-contained. The paper does not hide its limitations; the discussion after Theorem 5 even acknowledges the failure to get S(A) << |A|^{2/3}. Self-citation is fine here since Bloom's energy bound is a prior external theorem.

Who is this for? Anyone working on convex sets, additive energy, or Sidon sets. It would make a good reading-group paper for a combinatorial number theory group. The results deserve a serious referee, and with modest revision the paper would be in good shape. I would send it to peer review, flagging Theorem 6's convexity check as the point the referee needs to verify.

Referee Report

2 major / 4 minor

Summary. The paper studies several quantitative measures of additive structure in convex sets. Its main result (Theorem 2) is a construction of arbitrarily large convex sets A with T3(A) ≫ |A|^{3/2} non-trivial three-term arithmetic progressions. This is obtained from a more general construction (Theorem 6) of convex sets of size Θ(n) containing at least n/(4m) disjoint arithmetic progressions of length m. The paper also reproves Schoen's bound r_{A+A}(x) ≪ |A|^{2/3} and gives a construction showing this bound is optimal (Theorems 9, 15, 16), proves the existence of large Sidon subsets of convex sets (Theorem 4: S(A) ≫ |A|^{77/150-o(1)}), and constructs convex sets with small maximum Sidon subset (Theorem 5: S(A) ≪ |A|^{3/4}). A final section connects the representation-optimality construction to Jarník's theorem on lattice points on convex curves.

Significance. If the main construction is completed, the paper answers a natural and previously open question: convex sets can contain as many as |A|^{3/2} three-term arithmetic progressions, matching the conditional Cauchy-Schwarz upper bound up to a constant and providing a counterpoint to the heuristic that convex sets are additively unstructured. The optimality of Schoen's representation bound and the Sidon-set results are also substantial. The proofs are mostly self-contained and use standard tools; there is no numerical fitting or circular reuse of output. However, as written the core construction in Theorem 6 has a definitional gap and a uniformity issue, and the introduction contains an invalid contrapositive. These are correctable, but they require revision before the paper can be accepted.

major comments (2)

[Section 2, proof of Theorem 6] The block B_m is not defined. The text defines y_ℓ only for m ≤ ℓ < 2m, but then defines B_m = {f_m(k) : y_{m-1} ≤ k ≤ x_m}. Since y_{m-1} is never defined, the first block, and hence A = ∪_ℓ B_ℓ, is not well-defined. The subsequent verification of disjointness and convexity cannot be checked. This is load-bearing for Theorem 2. Please define y_{m-1} (or otherwise specify the first block) and re-verify the displayed inequalities for that choice.
[Section 2, proof of Theorem 6] The convexity of A is made to rest on the displayed inequalities claimed to hold for all m ≤ ℓ < 2m for 'sufficiently small a > 0', but no uniform smallness bound is proved. Since both x_ℓ and y_ℓ depend on ℓ, one needs a positive lower bound on min_ℓ min{1/(2(2x_ℓ-1)), 1/(2(x_ℓ^2-y_ℓ^2))}. The text only asserts that taking a small enough works; this is not a verification. The gap is likely repairable: using x_ℓ-y_ℓ ≈ n/ℓ and x_ℓ = O(n/m), the worst case is ℓ = m and gives a bound a ≪ m^2/n^2, so a = c m^2/n^2 with sufficiently small c appears to work. But as printed a reader cannot verify that the construction is sound.

minor comments (4)

[Section 1, after Eq. (2)] The sentence 'To put this observation in its contrapositive form, any construction with T3(A) ≥ |A|^{3/2} would also give a construction refuting Conjecture 1' is not a valid contrapositive. Conjecture 1 implies T3(A) ≪ |A|^{3/2+o(1)}; the contrapositive is that T3(A) failing to be O(|A|^{3/2+o(1)}) would refute the conjecture. A construction with T3(A) ≥ |A|^{3/2} is consistent with the conjecture because |A|^{3/2} ≤ |A|^{3/2+o(1)}. Please rephrase.
[Section 3, proof of Theorem 15] The text says 'for i ≤ m we add b_i - 1 points in the interval (x_{i+1}, x_i)'. Since x_{i+1} > x_i, this interval is empty; it should presumably be (x_i, x_{i+1}).
[Section 1, bullet list] In the summary bullet list, T(A) is used instead of T_3(A); please make the notation consistent.
[Section 5, proof of Theorem 4] The sentence 'The expected size of A is p|A|' should refer to A′ rather than A.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the constructions are explicit and the cited external theorems are independent.

full rationale

The paper's derivations are forward and non-circular. Theorem 6 constructs A explicitly via f_l(x)=ax^2+l(x+n), defines blocks B_l, and verifies convexity through stated gap inequalities; the parameter a is chosen as a sufficiently small constant of order m^2/n^2, not fitted to any target statistic, and the T3 count is deduced afterwards. Theorem 3 and Theorem 9 are proved from Erdos-Szekeres and Lemma 12 using only the definition of convexity. Theorem 4 uses Bloom's energy bound [2] and the paper's Theorem 3; Bloom [2] is an independent parameter-free prior theorem whose assumptions do not include the present results, so despite sharing an author it is legitimate external evidence and not circular. Theorem 5 uses Ruzsa-Zhelezov's theorem and the Kovari-Sos-Turan theorem, both external. Section 4 transparently presents Theorem 15 as a special case of Jarnik's classical construction rather than renaming a known result. I found two non-circular weaknesses, flagged here to follow the review rule: (1) In the proof of Theorem 6, B_l = {f_l(k): y_{l-1} <= k <= x_l} uses y_{m-1}, which is undefined as printed, and the uniform smallness of a is asserted ("The former is true for sufficiently small a>0, as taking a small enough ensures a(2x_l-1)<1/2 and a(x_l^2-y_l^2)<1/2, say. (In particular a can be taken to be of the order m^2/n^2.)") without a fully quantified uniform bound. This is a repairable completeness gap, not circularity: the worst case is l=m, and choosing a = c m^2/n^2 with small c works uniformly under n >= 8m^2; moreover a is not fitted to any prediction. (2) The introduction's claim that "any construction with T3(A) >= |A|^{3/2} would also give a construction refuting Conjecture 1" is an overstatement, since Cauchy-Schwarz only yields T3(A) <= |A|^{3/2+o(1)} under the energy conjecture; this is a logical error, not a circular dependency. No claimed prediction reduces by construction to an input, and no load-bearing argument rests on a self-citation chain.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No data are fitted in any empirical sense. The free parameters are construction constants chosen to make proofs work. The paper relies on standard background theorems and prior published results; no new entities, forces, particles, or dimensions are introduced.

free parameters (2)

a = a_{m,n} in f_ell(x) = sufficiently small, of order m^2/n^2
Introduced ad hoc in Theorem 6 to make the convex-gap inequalities hold; not fitted to any data.
p, the random subset probability in Theorem 4 = c |A|^{-73/150 - epsilon}
Chosen by hand to balance expected bad energy solutions against the size of the random subset; a proof device, not an empirical fit.

assumptions (5)

domain assumption Finite convex set is a sequence whose consecutive differences strictly increase, equivalently the graph of a strictly convex increasing function.
This definition frames all results and is stated at the start of Section 1.
standard math Erdos-Szekeres theorem: any sequence of length m contains a monotone subsequence of length floor(sqrt m).
Used in Corollary 11 to decompose gaps into many monotone subsequences for the proof of Theorem 9.
standard math Known energy upper bound E(A) << |A|^{123/50+o(1)} from Bloom [2].
Used as an input in the probabilistic proof of Theorem 4; it is a cited prior result by one of the authors.
standard math Ruzsa-Zhelezov theorem: for large n there exist B,C of size n such that B+C contains a convex set of size n^2.
Used as the engine for Theorem 5, the construction of convex sets with small Sidon number.
standard math Kovari-Sos-Turan bound for C4-free bipartite graphs.
Used in the proof of Theorem 5 to force a C4 in the graph of a large Sidon subset.

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Cite this review

Pith. "Pith review of Additive structure in convex sets." pith.science (2026). https://pith.science/paper/2VXPKDEP

@misc{pith2026250901568,
  author       = {Pith},
  title        = {Pith review of: Additive structure in convex sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2VXPKDEP}},
  note         = {Machine review of arXiv:2509.01568}
}

abstract

This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set $A$ containing $\Omega(|A|^{3/2})$ three-term arithmetic progressions.

Figures

Figures reproduced from arXiv: 2509.01568 by the authors.

**Figure 1.** An illustration showing how the construction of Theorem 15 can be derived from Jarn´ık’s construction of a convex curve with many lattice points. 5. Sidon sets in convex sets We now turn to the problem of finding large Sidon sets in convex sets. We first prove Theorem 4, using a standard probabilistic argument. Proof of Theorem 4. Let A′ ⊆ A be a p-random subset, where we include each x ∈ A independently with proba… view at source ↗

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Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Large convex sets in difference sets

Krishnendu Bhowmick, Ben Lund, and Oliver Roche-Newton. Large convex sets in difference sets. Mathematika, 70(3):Paper No. e12263, 16, 2024

work page 2024

[2] [2]

Control and its applications in additive combinatorics

Thomas Bloom. Control and its applications in additive combinatorics. arXiv:2501.09470, 2025

arXiv 2025

[3] [3]

P. Erd˝ os. Extremal problems in number theory, combinatorics and geometry. InProceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 51–70. PWN, Warsaw, 1984

work page 1983

[4] [4]

Erd¨ os and G

P. Erd¨ os and G. Szekeres. A combinatorial problem in geometry. Compositio Math., 2:463– 470, 1935

work page 1935

[5] [5]

M. Z. Garaev. On lower bounds for the L1-norm of some exponential sums. Mat. Zametki, 68(6):842–850, 2000

work page 2000

[6] [6]

Vojtˇ ech Jarn ´ ık.¨Uber die Gitterpunkte auf konvexen Curven. Math. Z., 24:500–518, 1926

work page 1926

[7] [7]

Koml´ os, M

J. Koml´ os, M. Sulyok, and E. Szemer´ edi. Linear problems in combinatorial number theory. Acta Math. Acad. Sci. Hungar., 26:113–121, 1975

work page 1975

[8] [8]

C. Pohoata. Sidon sets and sum-product phenomena. https://pohoatza.wordpress.com/ 2021/01/23/sidon-sets-and-sum-productphenomena/ , Jan 2021. Blog post

work page 2021