On an asymmetric additive energy inequality

Akshat Mudgal

On an asymmetric additive energy inequality

T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper establishes a purely combinatorial proof of the asymmetric additive energy inequality, replacing Fourier analysis with repeated Cauchy-Schwarz applications and a discrete midpoint-convexity lemma.

desk verdict A sound, honestly-framed short note: the real content is a Fourier-free proof of a known additive energy inequality via a clean discrete midpoint-convexity lemma; the key step checks out, and the note deserves a serious referee. read the letter →

arxiv 2607.25442 v1 pith:6ZAENYDD submitted 2026-07-28 math.NT math.CO

Akshat Mudgal This is my paper · ORCID

classification math.NTmath.CO MSC 11B30

keywords additiveenergyCauchy-SchwarzdiscretemidpointconvexityFourier-freeproofsumsetestimatesnon-abeliangroupsSchattennormssum-productphenomenon

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

The paper proves a general inequality bounding a mixed additive energy of 2d functions by the geometric mean of their self-energies. This bound was known via Fourier analysis; the point of the paper is that it has a proof using only Cauchy's inequality and a discrete midpoint-convexity condition on a finite simplex. The proof is significant because it extracts the combinatorial content of a tool widely used in additive combinatorics, and it comes with non-abelian and sumset variants.

What carries the argument

The central object is the discrete simplex H_d = {nonnegative integer d-tuples summing to d}. For each u in H_d, Γ_u counts weighted solutions to a system in which the functions w_i appear u_i times each. The pivotal step is the midpoint inequality Γ_{(u+v)/2}^2 ≤ Γ_u Γ_v for u,v with midpoint in H_d, obtained by Cauchy's inequality and relabeling. Setting f(u)=log Γ_u makes f midpoint convex, and Proposition 1.3, a discrete midpoint-to-global convexity extension, yields f(1,...,1) ≤ (1/d)∑ f(de_i), which is exactly the energy bound.

What would settle it

Take d=3, set G=Z, let ν be a single atom at 0, and choose w_i to be indicator functions of small intervals; compute Γ_{(1,1,1)}, Γ_{(2,1,0)}, and Γ_{(0,1,2)}. If Γ_{(1,1,1)}^2 > Γ_{(2,1,0)}Γ_{(0,1,2)} for some weights, the proof's pivotal midpoint claim (3.5) would be false. Alternatively, construct a function f on H_3 that satisfies every midpoint convexity inequality but violates f(1,1,1) ≤ (f(3,0,0)+f(0,3,0)+f(0,0,3))/3; that would refute Proposition 1.3.

Watch

Extended reading notes

Core claim

For an abelian group G and nonnegative finitely supported functions ν, w_1, ..., w_{2d}, the generalized additive energy E_{2d,ν}(w_1,...,w_{2d}) is at most the product of the individual self-energies E_{2d,ν}(w_i)^{1/(2d)}. The paper gives a proof that never passes to the dual group: it rewrites the energy as a weighted solution count, applies Cauchy's inequality in a way that splits the variables according to a vector u in the discrete simplex H_d, and then shows that the logarithms of these counts are midpoint convex. A discrete convexity extension lemma converts midpoint convexity into the desired global bound. The paper also records a non-abelian analogue proved by matrix trace and Scha

Load-bearing premise

The whole proof depends on the claim that after applying Cauchy's inequality to the weighted solution count Γ_z and relabelling the variables, the two resulting factors come out exactly as Γ_u and Γ_v; if that relabelling ever produced a weighted count other than Γ_u or Γ_v, the reduction to discrete midpoint convexity would collapse.

Editorial extensions

If this is right

The asymmetric energy inequality now has a Fourier-free proof, so the bound follows from purely combinatorial operations whenever such weighted solution counts can be defined.
A direct corollary: if each set A_i has self-energy E_{2d}(A_i) ≤ N^{2d-c}, then the sumset A_1+...+A_d has size at least N^c.
For arbitrary finite groups, the analogous inequality for S_{2d}(w_1,...,w_{2d}) holds, proved by expressing the energy as a trace and applying Schatten-norm inequalities.
The sumset theorem states that for any finite nonempty sets A_i in an abelian group, |A_1+...+A_d| ≥ (|dA_1|...|dA_d|)^{1/(2d)}.
For sets of real numbers, this combines with existing sum-product estimates to imply that at least one of the k-fold sumset or k-fold product set has size ≫_k N^{c'' (log k)^{1/8}}.

Reading between the lines

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

The discrete midpoint-convexity lemma is stated for the simplex H_d, but its proof mechanism—moving mass between coordinates and iterating midpoint convexity—likely extends to other finite convex subsets of Z^d, potentially yielding analogous multilinear inequalities for other energy functionals.
A natural quantitative test is to compute the ratio Γ_{(u+v)/2}^2 / (Γ_u Γ_v) for small d and random weights; if the ratio is typically close to 1, the argument may admit sharpened or almost-sharp versions rather than a purely qualitative bound.
The gap between the abelian combinatorial proof and the non-abelian spectral proof suggests a structural boundary: the paper leaves open whether a Fourier-free proof exists for non-abelian groups, and that question could clarify the true role of commutativity in these energy inequalities.

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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 thing to know: the main inequality is known. Theorem 1.1 is the standard asymmetric additive energy bound, and §2 gives the usual Fourier proof. The actual contribution is §3 — a purely combinatorial proof via repeated Cauchy–Schwarz plus a discrete midpoint-convexity lemma (Proposition 1.3). I checked the critical step, the log-midpoint convexity of Γ_u, including the relabeling after Cauchy. It works: the block-by-block re-enumeration gives exactly Γ_u and Γ_v, the h_i = 1 swap does its job, and the reduction to Proposition 1.3 is sound. I don't see a load-bearing flaw.

Credit where due. The paper is unusually honest about scope. Theorem 1.1 is flagged as known, Theorem 1.4 is explicitly a special case of Hatami's graph-norm work, and Proposition 1.3 is claimed only in the weak sense that the author could not find it in the discrete-convexity literature. That is the right framing. Proposition 1.3 looks like the most valuable piece — a clean statement with a short, correct proof that could find use elsewhere. The sumset result (Theorem 1.5) is a nice one-page application of Plünnecke–Ruzsa, and Corollary 1.6 is a modest but real sum-product consequence of the author's earlier work plus Gowers–Green–Manners–Tao.

Soft spots, in proportion. Significance is genuinely modest: nothing here changes a theorem, and the Fourier proof is shorter than the combinatorial one. The value is in the toolkit, not the result. The novelty claim for Proposition 1.3 is absence-of-literature evidence, not a substantive search; the author says as much, and a referee should weigh it accordingly. The relabeling in (3.5) is terse — reconstructable and correct, but a referee will want it spelled out. The self-citations in Corollary 1.6 are used as established results, not circularly. The Copilot disclosure in the acknowledgements is transparent, and the proof is checkable regardless of provenance.

Who it's for: people in additive combinatorics who care about Fourier-free arguments, and anyone using this energy inequality in Waring, Vinogradov, or sum-product contexts. It deserves a serious referee and minor revision, not a desk reject.

Referee Report

0 major / 4 minor

Summary. The paper proves the generalized additive energy inequality $E_{2d,\nu}(w_1,\dots,w_{2d}) \le \prod_i E_{2d,\nu}(w_i)^{1/2d}$ for abelian groups, first by a short Fourier analytic argument and then by a purely combinatorial proof. The combinatorial proof reduces to a discrete midpoint-convexity result (Proposition 1.3) via repeated Cauchy–Schwarz. The paper also contains a non-abelian analogue (Theorem 1.4) using Schatten norms and a sumset lower bound (Theorem 1.5) derived from the Plünnecke–Ruzsa inequality, with an application to real sum-product estimates (Corollary 1.6).

Significance. The main inequality is not new, but the combinatorial proof is a novel contribution; it avoids Fourier/spectral analysis and introduces a clean discrete convexity lemma that may be useful elsewhere. The non-abelian and sumset results are correct and well-motivated. The proofs are self-contained and line-by-line checkable. The paper is appropriately concise for a note.

minor comments (4)

[§2, Eq. (2.1)] The displayed Fourier identity is garbled in the typeset version: the hats on w_{d+1},...,w_{2d} appear to be missing, and the exponent of |ν̂| is unclear (should be |ν̂|^2, with the extra '2d' belonging to the measure dμ*). Please correct.
[§3, proof of (3.5)] The relabeling step after the Cauchy–Schwarz application is very terse. It would help to state explicitly that the kept variables are re-enumerated block by block according to the cumulative sums of u (and v), and that when h_i=1 the pair (a_i,a'_i) is swapped so that every term has the form x_i-y_i. As written, this is the only place where a careful reader may stumble.
[§5 / Corollary 1.6] The exponents in (1.7) and (1.8) are ambiguous in the rendering, e.g. 'N c′(logk) 1/8' should be $N^{c'(\log k)^{1/8}}$. Also, the multiplicative application implicitly passes to the group of nonzero reals; a remark on how zero/negative elements are handled would be useful.
[§4] In (4.3), the identity Λ(w_1,...,w_{2d}) = |G| S_{2d}(w_1,...,w_{2d}) is used without stating the factor |G| explicitly; this might confuse readers. Consider writing Λ = |G| S in a displayed line.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the combinatorial proof of Theorem 1.1 is self-contained.

full rationale

The central derivation chain for Theorem 1.1 is the combinatorial proof in §3. It uses only Cauchy–Schwarz, double-counting, the definition of the auxiliary quantities Γ_u, and the discrete midpoint-convexity Proposition 1.3. Proposition 1.3 is proved independently in §3 by an elementary max-support-decrease argument; it is not assumed from prior work and does not presuppose the target inequality. The midpoint log-convexity claim (3.5) is established directly by applying Cauchy's inequality and then relabelling the variables; it is not a restatement of the desired bound. The Fourier proof in §2 is also standard and self-contained. The only self-citations appear in the applications and remarks around Corollary 1.6, where [11], [12], and [13] are cited as established external results for sum-product estimates; these are not load-bearing inputs to the main theorem and do not reduce any central claim to a self-citation. The note that Copilot was used in proving Proposition 1.3 does not create a circular dependency. No fitted input is relabelled as a prediction, and no known result is merely renamed. Thus the paper exhibits no significant circularity.

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

No free parameters or invented entities; the paper is a pure-math proof. It relies on standard analytic inequalities and on external additive combinatorics results for the secondary corollaries; the central combinatorial proof depends only on Cauchy–Schwarz and the proved Proposition 1.3.

assumptions (5)

standard math Cauchy–Schwarz inequality
Used throughout §3 (e.g., (3.1)) to split the additive energy and in the midpoint-convexity proof of Γ.
standard math Hölder's inequality and Fourier orthogonality on finitely generated abelian groups
Used in §2 for the standard Fourier proof of Theorem 1.1; the combinatorial proof avoids these, but they are invoked to establish the known result.
standard math von Neumann trace inequality and Hölder for Schatten norms
Used in §4 to prove the non-abelian Theorem 1.4 via trace bounds.
domain assumption Plünnecke–Ruzsa inequality (Lemma 5.1, [21, Cor. 6.28])
External additive-combinatorics theorem used to prove Theorem 1.5; not proved in the paper.
domain assumption Sum-product estimates from [5] and author's prior work [11],[12] (inequality (1.7))
Used only for Corollary 1.6; the central Theorem 1.1 proof does not depend on these.

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Cite this review

Pith. "Pith review of On an asymmetric additive energy inequality." pith.science (2026). https://pith.science/paper/6ZAENYDD

@misc{pith2026260725442,
  author       = {Pith},
  title        = {Pith review of: On an asymmetric additive energy inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ZAENYDD}},
  note         = {Machine review of arXiv:2607.25442}
}

abstract

Let $d \geq 1$ be an integer, $G$ be an abelian group and $\nu, w_1, \dots, w_{2d}: G \to [0, \infty)$ be functions with finite, non-empty supports. Define the generalised additive energy \[ E_{2d, \nu}(w_1, \dots, w_{2d}) = \sum_{y,y' \in G}\sum_{a_1, \dots, a_{2d} \in G } w_1(a_1) \dots w_{2d}(a_{2d}) \nu(y) \nu(y') 1_{\sum_{i=1}^d (a_i - a_{i+d}) = y-y'} .\] Moreover, for every $1 \leq i \leq 2d$, let $E_{2d, \nu}(w_i) = E_{2d, \nu}(w_i, \dots, w_i)$. A standard Fourier analytic argument delivers the estimate \[ E_{2d,\nu}(w_1, \dots, w_{2d}) \leq \prod_{1 \leq i \leq 2d} E_{2d, \nu}(w_i)^{1/2d}.\] In this note, we present a purely combinatorial proof of the above inequality. In particular, our proof does not use any Fourier or spectral analysis and relies on repeated applications of Cauchy--Schwarz inequality combined with a discrete convexity extension type argument. We also record a variation of this upper bound in the non-abelian setting via spectral inequalities following work of Hatami on graph norms, as well as a relevant sumset analogue obtained via iterative applications of the Pl\"{u}nnecke--Ruzsa inequality.

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Works this paper leans on

24 extracted references · 2 linked inside Pith

[1]

T. F. Bloom, W. Sawin, C. Schildkraut, D. Zhelezov,The sum-product conjecture is false for real numbers, arXiv:2605.28781

arXiv
[2]

Bourgain,Estimates on exponential sums related to the Diffie-Hellman distributions, Geom

J. Bourgain,Estimates on exponential sums related to the Diffie-Hellman distributions, Geom. Funct. Anal.15(2005), no. 1, 1–34

2005
[3]

Bourgain, M.-C

J. Bourgain, M.-C. Chang,On the size ofk-fold sum and product sets of integers, J. Amer. Math. Soc. 17(2004), no. 2, 473–497

2004
[4]

Chang,The Erd˝ os-Szemer´ edi problem on sum set and product set, Ann

M.-C. Chang,The Erd˝ os-Szemer´ edi problem on sum set and product set, Ann. of Math. (2)157(2003), no. 3, 939–957

2003
[5]

W. T. Gowers, B. Green, F. Manners, T. Tao,On a conjecture of Marton, Ann. of Math. (2)201 (2025), no. 2, 515–549

2025
[6]

L. Guth, J. Maynard,New large value estimates for Dirichlet polynomials, Ann. of Math. (2)203 (2026), no. 2, 623–675

2026
[7]

Hanson, O

B. Hanson, O. Roche-Newton, D. Zhelezov,On iterated product sets with shifts,II, Algebra Number Theory14(2020), no. 8, 2239–2260

2020
[8]

Harrison, A

J. Harrison, A. Mudgal, H. Schmidt,Uniform sum-product phenomenon for algebraic groups and Brem- ner’s conjecture, arXiv:2603.06483

arXiv

Show all 24 references

[9]

Hatami,Graph norms and Sidorenko’s conjecture, Israel J

H. Hatami,Graph norms and Sidorenko’s conjecture, Israel J. Math.175(2010), 125–150

2010
[10]

Moriguchi, K

S. Moriguchi, K. Murota, A. Tamura, F. Tardella,Discrete midpoint convexity, Math. Oper. Res.45 (2020), no. 1, 99–128

2020
[11]

Mudgal,An Elekes–R´ onyai theorem for sets with few products, Int

A. Mudgal,An Elekes–R´ onyai theorem for sets with few products, Int. Math. Res. Not. IMRN (2024), no. 13, 10410–10424

2024
[12]

Mudgal,Energy estimates in sum-product and convexity problems, Amer

A. Mudgal,Energy estimates in sum-product and convexity problems, Amer. J. Math.147(2025), no. 4, 869–899

2025
[13]

Mudgal,Unbounded expansion of polynomials and products, Math

A. Mudgal,Unbounded expansion of polynomials and products, Math. Ann.390(2024), no. 1, 381–415

2024
[14]

Mudgal,On commuting pairs in arbitrary sets of2×2matrices, to appear in Mathematische Zeitschrift, arXiv:2411.10404

A. Mudgal,On commuting pairs in arbitrary sets of2×2matrices, to appear in Mathematische Zeitschrift, arXiv:2411.10404

arXiv
[15]

P´ alv¨ olgyi, D

D. P´ alv¨ olgyi, D. Zhelezov,Query complexity and the polynomial Freiman-Ruzsa conjecture, Adv. Math. 392(2021), Paper No. 108043, 18 pp

2021
[16]

Petridis,New proofs of Pl¨ unnecke-type estimates for product sets in groups, Combinatorica32 (2012), no

G. Petridis,New proofs of Pl¨ unnecke-type estimates for product sets in groups, Combinatorica32 (2012), no. 6, 721–733

2012
[17]

I. Z. Ruzsa,On the number of sums and differences, Acta Math. Hungar.59(1992), no. 3–4, 439–447

1992
[18]

I. D. Shkredov,Some new inequalities in additive combinatorics, Mosc. J. Comb. Number Theory3 (2013), no. 3–4, 189–239

2013
[19]

I. D. Shkredov,Some new results on higher energies, Trans. Moscow Math. Soc.2013, 31–63

2013
[20]

I. D. Shkredov,Modular hyperbolas and bilinear forms of Kloosterman sums, J. Number Theory220 (2021), 182–211

2021
[21]

T. Tao, V. Vu,Additive combinatorics, Cambridge Stud. Adv. Math.,105, Cambridge University Press, Cambridge, 2006. xviii+512 pp. 12 AKSHAT MUDGAL

2006
[22]

T. D. Wooley,Large improvements in Waring’s problem, Ann. of Math. (2)135(1992), no. 1, 131–164

1992
[23]

T. D. Wooley,The cubic case of the main conjecture in Vinogradov’s mean value theorem, Adv. Math. 294(2016), 532–561

2016
[24]

T. D. Wooley,Nested efficient congruencing and relatives of Vinogradov’s mean value theorem, Proc. Lond. Math. Soc. (3)118(2019), no. 4, 942–1016. Mathematics Institute, Zeeman Building, University of W arwick, Coventry CV4 7AL, UK Email address:Akshat.Mudgal@warwick.ac.uk

2019

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

T. F. Bloom, W. Sawin, C. Schildkraut, D. Zhelezov,The sum-product conjecture is false for real numbers, arXiv:2605.28781

arXiv

[2] [2]

Bourgain,Estimates on exponential sums related to the Diffie-Hellman distributions, Geom

J. Bourgain,Estimates on exponential sums related to the Diffie-Hellman distributions, Geom. Funct. Anal.15(2005), no. 1, 1–34

2005

[3] [3]

Bourgain, M.-C

J. Bourgain, M.-C. Chang,On the size ofk-fold sum and product sets of integers, J. Amer. Math. Soc. 17(2004), no. 2, 473–497

2004

[4] [4]

Chang,The Erd˝ os-Szemer´ edi problem on sum set and product set, Ann

M.-C. Chang,The Erd˝ os-Szemer´ edi problem on sum set and product set, Ann. of Math. (2)157(2003), no. 3, 939–957

2003

[5] [5]

W. T. Gowers, B. Green, F. Manners, T. Tao,On a conjecture of Marton, Ann. of Math. (2)201 (2025), no. 2, 515–549

2025

[6] [6]

L. Guth, J. Maynard,New large value estimates for Dirichlet polynomials, Ann. of Math. (2)203 (2026), no. 2, 623–675

2026

[7] [7]

Hanson, O

B. Hanson, O. Roche-Newton, D. Zhelezov,On iterated product sets with shifts,II, Algebra Number Theory14(2020), no. 8, 2239–2260

2020

[8] [8]

Harrison, A

J. Harrison, A. Mudgal, H. Schmidt,Uniform sum-product phenomenon for algebraic groups and Brem- ner’s conjecture, arXiv:2603.06483

arXiv