Entropic additive energy and entropy inequalities for sums and products

Ioannis Kontoyiannis; Lampros Gavalakis; Rupert Li

REVIEW 2 major objections 2 minor 27 references

Entropic additive energy and entropy inequalities for sums and products

T0 review · 2 major / 2 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The entropy of any sum of products is bounded by marginal entropy plus doubling constants.

desk verdict The ring Plünnecke–Ruzsa inequality is genuine and the proofs survive a line-by-line audit; the main defects are typos, not math, so send it out. read the letter →

arxiv 2506.20813 v2 pith:47RH3N3G submitted 2025-06-25 cs.IT math.COmath.ITmath.PR

Rupert Li , Lampros Gavalakis , Ioannis Kontoyiannis This is my paper · ORCID

classification cs.ITmath.COmath.ITmath.PR MSC 94A1711B3011P70

keywords differentialentropyadditiveenergyringPlünnecke–Ruzsainequalitysum-productinequalitiesmultiplicativeBalog–Szemerédi–GowerstheorementropicdoublingconstantErdős–Szemerédiphenomenon

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

This paper proves new entropy inequalities for sums, products, and sum-product combinations of continuous random variables, extending the dictionary between additive combinatorics and information theory. The central result is a ring Plünnecke–Ruzsa inequality: the differential entropy of a sum of products of i.i.d. copies of $X$ is bounded by the entropy of one product term plus a combination of $X$'s additive and multiplicative doubling constants. The paper also introduces a continuous analog of additive energy and shows, with exact mutual-information corrections, that large additive energy is equivalent to small entropy of the sum; it proves a Balog–Szemerédi–Gowers theorem for differential entropy and a discrete stability theorem showing that near-maximal doubling forces approximate support on a Sidon set (a set whose pairwise sums are all distinct). Finally, it shows that an entropic Erdős–Szemerédi phenomenon for integer-valued variables, if it holds at all, can involve an exponent of at most $4/3$, in contrast with known combinatorial bounds. Together these are structural bounds: complicated sum-product entropy is controlled by pairwise doubling statistics, with no distributional assumptions beyond finite entropy and integrable log-magnitude.

What carries the argument

The load-bearing objects are the differential entropic additive energy $a(X,Y)=2h(X,Y)-h(X+Y)$, the multiplicative entropy $\tilde h(X)=h(X)-E[\log|X|]$, and the additive and multiplicative doubling constants $\sigma(X)=h(X+X')-h(X)$, $\tilde\sigma(X)=h(XX')-h(X)-E[\log|X|]$, and $\delta(X)=h(X-X')-h(X)$. The proofs replace the functional submodularity of discrete entropy, which fails for differential entropy, with the data processing inequality for mutual information; this supplies the intermediate inequalities from which the ring Plünnecke–Ruzsa bounds follow inductively.

What would settle it

Take a continuous $X$ with finite entropy and $E[\log|X|]$, for instance a log-normal or uniform-on-$[1,N]$ variable, compute $h(X_{1,1}X_{1,2}-X_{2,1}X_{2,2})$ numerically together with $\sigma(X)$ and $\tilde\sigma(X)$, and check whether it exceeds $h(X_1X_2)+4\tilde\sigma(X)+2\sigma(X)$; any violation at these $m=n=2$ parameters would refute the ring inequality.

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Extended reading notes

Core claim

The paper's central discovery is a ring Plünnecke–Ruzsa inequality for differential entropy (Theorem 7.6): for an i.i.d. array $\{X_{i,j}:1\le i\le m,\,1\le j\le n\}$ distributed as $X$, $$h\!\left(\sum_{i=1}^m\prod_{j=1}^n X_{i,j}\right)\le h(X_{1,1}\cdots X_{1,n}) +(m-1)\bigl[(n+2)(n-1)\tilde\$\sigma$(X)+(n-1)\delta(X)+\$\sigma$(X)\bigr],$$ where $\sigma(X)=h(X+X')-h(X)$, $\delta(X)=h(X-X')-h(X)$, and $\tilde\sigma(X)=h(XX')-h(X)-E[\log|X|]$ for an independent copy $X'$. A discrete analogue (Theorem 7.7) holds for discrete random variables taking values in an arbitrary integral domain. The inequality says that the entropy of an arbitrary sum of products is governed by the marginal entropy of one product term plus the additive and multiplicative doubling constants of $X$, with no further structural assumptions. The proof is built from a continuous version of additive energy, multiplicative analogues of the Ruzsa and submodularity inequalities, and repeated uses of the data processing inequality.

Load-bearing premise

The multiplicative results require $\log|X|$ to be integrable and all differential and multiplicative entropies to be finite, and the proof imports the additive and multiplicative Plünnecke–Ruzsa inequalities as black boxes; if any of these fail, the bounds either degenerate or lose their support.

Editorial extensions

If this is right

For $m$-term sums of $n$-fold products of i.i.d. $X$, the entropy penalty beyond the marginal term grows at most linearly in $m$ and quadratically in $n$, with coefficients fixed by $\sigma(X)$, $\delta(X)$, and $\tilde\sigma(X)$.
When both additive and multiplicative doubling are bounded by $\log K$, the bound becomes $H(\sum_i\prod_j X_{ij})\le H(X)+[(n-1)+(m-1)(n^2+3n-3)]\log K$, matching a discrete bound obtained independently in recent work cited in the paper.
Large differential additive energy is quantitatively equivalent to small entropy of the sum, exactly through the mutual-information correction $2I(X;Y)$; this makes the 'energy large iff sum entropy small' heuristic a theorem for continuous variables.
The differential Balog–Szemerédi–Gowers theorem yields conditionally independent $X_1,Y_2$ given $X+Y$ with $h(X_1+Y_2|X+Y)$ small, so large additive energy can be converted into an explicit small-doubling structure inside the joint distribution.
For discrete variables, near-maximal doubling $H(X+X')$ implies approximate support on a Sidon set, extending inverse sumset theory from the small-doubling to the large-doubling regime.

Reading between the lines

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

The $m$-linear, $n$-quadratic form of the bound suggests that entropy of sums of products can be controlled by pairwise statistics alone; an analogous bound for arrays with dependencies among the $X_{i,j}$ would likely need new mutual-information terms, and this is the first natural extension to test.
The continuous BSG theorem and the $\epsilon\le 1/3$ obstruction together suggest that entropic sum-product phenomena behave differently from set cardinality: the entropy of a sum of two uniform draws can stay near $\log n$ even when the sumset is large, so any transfer from combinatorial to entropic results must go through distributional conditioning rather than support-size arguments.
One concrete next test is computational: for a family of densities (log-normal, exponential, uniform on $[1,N]$, and mixtures with atoms at 0), estimate the left- and right-hand sides of Theorem 7.2 for $n=2$; the predicted tightness or slack as $\tilde\sigma(X)$ and $\sigma(X)$ vary would guide which distributions, if any, saturate the bound.

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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

Bottom line: this is a solid paper and the main new results are real. The continuous additive energy definition and the representation a(X,Y)=2h(X,Y)-h(X+Y) make the "large energy iff small sum entropy" story clean, and the differential BSG theorem (Thm 4.3) improves on the prior bound in the main inequality. The general ring Plünnecke–Ruzsa inequality (Thms 7.6–7.7) is the centerpiece. I re-derived the induction in Prop 7.4 and the passage through Cor 7.5; the h(X) coefficients cancel and the constants match. The discrete version being identical to the updated Mathé–O'Regan preprint is acknowledged, which is good practice.

Section 8's counterexample is the right negative result: it shows the naive entropic Erdős–Szemerédi conjecture can hold only with epsilon ≤ 1/3, and the bookkeeping with the point mass at 0 is honest.

The soft spots are minor. The sum-difference inequality is printed twice with 3h(X+Y) instead of 3h(X−Y) (Sections 1.2 and 2.3). That's confusing but the proof and the Ruzsa-distance form make the intended statement unambiguous. There is also a compressed intermediate step in the proof of Theorem 7.2 where Lemma 7.3 is applied; the stress-test re-derivation checks out, but a referee will want that line expanded. The paper relies on the earlier Plünnecke–Ruzsa inequalities from [12] and [16] as black boxes; those are published, parameter-free results, so this is not a circularity. The integrability and finiteness assumptions are stated where used.

One point I'd push back on regarding the reader's report: the typographical errors are not structural. The verdict should be accept with minor revisions, not a conditional that suggests the main argument might fail. The math is in good shape.

This paper is for specialists in entropic additive combinatorics and anyone applying sumset methods in information theory. It deserves a serious referee. I'd send it out, with a request to fix the typos and expand the compressed proof step.

Referee Report

2 major / 2 minor

Summary. This paper develops differential-entropy analogues of additive-combinatorial inequalities. It introduces a continuous additive energy a(X,Y)=2h(X,Y)-h(X+Y), proves several large-energy/small-sum equivalences and a Balog-Szemerédi-Gowers theorem, gives a stability result for discrete variables with large doubling, translates additive sumset inequalities to products via multiplicative entropy, and proves a ring Plünnecke-Ruzsa inequality bounding the entropy of sums of products of i.i.d. variables by the marginal entropy plus additive and multiplicative doubling constants. A final section gives an example showing that an entropic Erdős-Szemerédi sum-product phenomenon, if true, can hold only with a restricted range of parameters.

Significance. The central Theorem 7.6 is a genuinely new structural bound in the Plünnecke-Ruzsa spirit, with explicit constants and no fitted parameters. The proofs are mostly clean chain-rule and data-processing arguments; I verified the induction in Proposition 7.4 and the applications of the additive and multiplicative Plünnecke-Ruzsa inequalities. The paper is therefore a potentially valuable contribution to the entropy/additive-combinatorics literature. Its strengths include the parameter-free derivations, the exact constants, and the honest negative result in Section 8. However, the paper also contains two incorrect equivalences in Section 3 that must be fixed before the advertised claims about additive energy and entropy of sums are valid.

major comments (2)

[Section 3, Corollary 3.5, Eqs. (31)-(34)] The equivalence is mis-stated. The proof's own calculation gives, after substituting h(X+Y)=h(X|Y)+I(X+Y;Y) and h(Y|X)=h(Y)-I(X;Y), the inequality (1/2)h(X)-(1/2)h(Y)-log C + I(X+Y;Y)+I(X;Y) <= 0, which rearranges to h(X) <= h(Y)+2 log C - 2 I(X+Y;Y) - 2 I(X;Y). The printed Eq. (31) has +2I(X;Y). With the printed sign, the implication (31) => (30) fails: for X~N(0,1), Y=X+N, N~N(0,0.1^2), and C=10, the printed (31) holds, but a(X,Y) is about -1.04, which is not >= (3/2)h(X)+(3/2)h(Y)-log 10 (about 1.96). The same sign error propagates to (32) and (33), and (34) should be correspondingly strengthened after the correction.
[Section 3, Corollary 3.7] The equivalence has the wrong sign in front of log C. From a(X,Y)=2h(X,Y)-h(X+Y), the condition h(X+Y) >= h(X)+h(Y)+log C implies a(X,Y) <= h(X)+h(Y)-log C - 2I(X;Y), not <= h(X)+h(Y)+log C - 2I(X;Y) as printed. The printed version is false: for X,Y i.i.d. N(0,1) and C=e^{-1.1}, the left-hand inequality holds, but the printed right-hand inequality fails, since a(X,Y) is about 3h(X)-0.35 while 2h(X)-1.1 is much smaller.

minor comments (2)

[Section 1.2] The displayed continuous sum-difference inequality should be h(X+Y)+h(X)+h(Y) <= 3h(X-Y); as printed, the right-hand side is 3h(X+Y). This is a typo only: the proof in Section 2.3 uses the correct statement d(X,-Y) <= 3d(X,Y).
[Theorem 7.6] The passage from Corollary 7.5 to the m-fold sum is summarized in one sentence. The step is a legitimate induction on m using the additive Plünnecke-Ruzsa inequality with K_i = exp(C), but it would be clearer to spell out that the products P_i are independent and satisfy the pairwise bound h(P_1+P_i) <= h(P_1)+C.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central ring Plünnecke–Ruzsa inequality follows from a direct induction and previously established, parameter-free inequalities; self-citations are not load-bearing in a circular sense.

full rationale

The main derivation chain (Proposition 7.4 → Corollary 7.5 → Theorems 7.6 and 7.7) does not reduce to its own inputs. Proposition 7.4 is an induction whose base case is Theorem 7.2, and the proof algebraically cancels the h(A), h(AX_n), and E[log|X|] terms to produce exactly the claimed coefficients 3, 2, (n−1), and −(n+2)(n−1). Corollary 7.5 then applies the multiplicative Plünnecke–Ruzsa inequality (Theorem 6.6), which is a stated result from prior published work [16] with explicit integrability assumptions and no target sum-product conclusion; this is legitimate independent support rather than circular self-citation. The final step applies the additive Plünnecke–Ruzsa inequality from [12] to independent products with the relevant K_i, again a parameter-free published lemma whose assumptions do not include the target result. The discrete Theorem 7.7 is explicitly compared with the independently obtained recent result of Mathé and O'Regan, confirming rather than renaming it. The Section 3 relation a(X,Y)=2h(X,Y)−h(X+Y) is a definition, and the 'large additive energy iff small sum entropy' corollaries are transparent algebraic equivalences following from that definition; they are not disguised empirical predictions, fitted constants, or post-hoc exclusions. No fitted parameters, invented constants, or externally imported uniqueness claims appear in the proof chain. A typo in the introductory sum-difference inequality display (h(X+Y)+h(X)+h(Y)≤3h(X+Y) in place of 3h(X−Y)) is immaterial to the main results, since the actual proof derives d(X,−Y)≤3d(X,Y).

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities appear. The axioms are regularity assumptions (finite entropies, integrable log|X|) and established prior theorems used as black boxes. None of these assumptions are tailored to force the target results.

assumptions (6)

domain assumption All differential entropies appearing are assumed to exist and be finite.
Stated in Section 2.3 and Section 6; used throughout for chain rules and conditioning.
domain assumption log|X| is integrable for multiplicative entropy results.
Stated before Lemma 6.1, Proposition 6.3, and Theorem 7.6; needed for E[log|X|] terms.
standard math Additive Plünnecke-Ruzsa inequality (23) from [12] holds for differential entropy.
Imported as a black box in deriving Corollary 7.5 and Theorem 7.6.
standard math Multiplicative Plünnecke-Ruzsa inequality (Theorem 6.6) follows from [16] and is used as a black box.
Used to pass from Proposition 7.4 to Corollary 7.5 and Theorem 7.6.
standard math Data processing inequality for mutual information, including Markov chains of the form X -> (XY, XZ) -> X(Y+Z), where the last variable is a deterministic function of the second.
Used in Proposition 7.1 and Lemma 7.3; correctness verified in this review.
standard math Doubling-difference inequality sigma(X) and delta(X) for differential entropy, i.e., 1/2 <= sigma(X)/delta(X) <= 2.
Imported from [12] and used in Theorem 7.2 and Theorem 7.7.

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Cite this review

Pith. "Pith review of Entropic additive energy and entropy inequalities for sums and products." pith.science (2026). https://pith.science/paper/47RH3N3G

@misc{pith2026250620813,
  author       = {Pith},
  title        = {Pith review of: Entropic additive energy and entropy inequalities for sums and products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/47RH3N3G}},
  note         = {Machine review of arXiv:2506.20813}
}

read the original abstract

Following a growing number of studies that, over the past 15 years, have established entropy inequalities via ideas and tools from additive combinatorics, in this work we obtain a number of new bounds for the differential entropy of sums, products, and sum-product combinations of continuous random variables. Partly motivated by recent work by Goh on the discrete entropic version of the notion of "additive energy", we introduce the additive energy of pairs of continuous random variables and prove various versions of the statement that "the additive energy is large if and only if the entropy of the sum is small", along with a version of the Balog-Szemer\'edi-Gowers theorem for differential entropy. Then, motivated in part by recent work by M\'ath\'e and O'Regan, we establish a series of new differential entropy inequalities for products and sum-product combinations of continuous random variables. In particular, we prove a new, general, ring Pl\"unnecke-Ruzsa entropy inequality. We briefly return to the case of discrete entropy and provide a characterization of discrete random variables with "large doubling", analogous to Tao's Freiman-type inverse sumset theory for the case of small doubling. Finally, we consider the natural entropic analog of the Erd\"os-Szemer\'edi sum-product phenomenon for integer-valued random variables. We show that, if it does hold, then the range of parameters for which it does would necessarily be significantly more restricted than its anticipated combinatorial counterpart.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [12]

Kontoyiannis and M

I. Kontoyiannis and M. Madiman. Sumset and inverse sumset inequalities for differential entropy and mutual information.IEEE Trans. Inform. Theory, 60(8):4503–4514, August 2014

work page 2014

[2] [16]

Madiman and I

M. Madiman and I. Kontoyiannis. Entropy bounds on abelian groups and the Ruzsa divergence.IEEE Trans. Inform. Theory, 64(1):77–92, January 2018

work page 2018

[3] [1]

Balog and E

A. Balog and E. Szemer´ edi. A statistical theorem of set addition.Combinatorica, 14(3):263–268, September 1994

work page 1994

[4] [2]

T.F. Bloom. Control and its applications in additive combinatorics.arXiv e-prints,2501.09470 [math.NT], January 2025

arXiv 2025

[5] [3]

Erd˝ os and E

P. Erd˝ os and E. Szemer´ edi. On sums and products of integers. In P. Erd˝ os, L. Alp´ ar, G. Hal´ asz, and A. S´ ark¨ ozy, editors,Studies in pure mathematics, pages 213–218. Birkh¨ auser, Basel, 1983

work page 1983

[6] [4]

Gavalakis, I

L. Gavalakis, I. Kontoyiannis, and M. Madiman. The entropic doubling constant and robustness of Gaussian codebooks for additive-noise channels.IEEE Trans. Inform. Theory, 70(12):8467–8477, De- cember 2024. 25

work page 2024

[7] [5]

M. Goh. On an entropic analogue of additive energy.arXiv e-prints,2406.18798 [math.CO], June 2024

arXiv 2024

[8] [6]

W.T. Gowers. A new proof of Szemer´ edi’s theorem for arithmetic progressions of length four.Geom. Funct. Anal., 8(3):529–551, July 1998

work page 1998