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This paper proves the entropic sum-product phenomenon: for any i.i.d. discrete real-valued random variables with finite entropy, either the sum or the product has entropy at least 8/7 of the original entropy, up to a logarithmic correction.

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2026-08-03 14:44 UTC pith:YFR32NW6

load-bearing objection First real progress on Goh's entropic sum-product conjecture: δ=1/7 with a long but sound proof.

arxiv 2607.29042 v1 pith:YFR32NW6 submitted 2026-07-31 cs.IT math.COmath.ITmath.PR

The Entropic Sum-Product Phenomenon

Rupert Li This is my paper
classification cs.IT math.COmath.ITmath.PR MSC 94A1711B7560E05
keywords entropic sum-product phenomenonShannon entropysum-product conjecturepoint-plane incidence bounddyadic uniformizationmultiplicative energyfew-sums-many-productsadditive doubling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves the entropic analog of the sum-product phenomenon: for any two independent copies of a discrete real-valued random variable with finite Shannon entropy, either their sum or their product must have entropy at least 8/7 of the original entropy, up to an O(log H) correction. This answers a conjecture in the literature that merely asked for a coefficient strictly larger than 1. A sympathetic reader should care because it shows a fundamental tension: a probability distribution cannot be nearly closed under both addition and multiplication, just as a set of numbers cannot. The previous tools were blocked by distributions with a large gap between Shannon entropy and min-entropy; the paper removes this blockage by slicing the distribution into near-uniform pieces, at a cost of only O(log H) bits. Two complementary linear bounds are then combined, one effective at small additive doubling and one at large doubling, yielding the 8/7 coefficient.

Core claim

Central claim: max{H(X+X'), H(XX')} ≥ (8/7)H(X) − O(log H(X)) for any i.i.d. discrete real-valued variables with finite entropy; asymptotically this is (1 + 1/7 − o(1))H(X). The proof cuts the law into near-uniform dyadic pieces at O(log H) cost, proving two zero-free linear bounds: product entropy ≥ (2 − 6eu − o(1))H and ≥ (5/4 − 3eu/4 − o(1))H, where eu is relative additive doubling. Their envelope crosses at eu = 1/7; a recombining step preserves the coefficient when there is an atom at zero, and truncation covers all discrete laws.

What carries the argument

Three tools carry the argument. A dyadic uniformization lemma cuts any finite-entropy law into near-uniform pieces of size powers of two, at a total loss of O(log H) bits, replacing the min-entropy bottleneck of earlier work. Two linear entropic few-sums-many-products bounds are then proved: one from a multiplicative-energy estimate on sign-pure sets, giving product entropy ≥ 2 − 6eu; the other from a point-plane incidence bound, giving product entropy ≥ 5/4 − 3eu/4. Finally, a recombination proposition converts the envelope of these two zero-free bounds into a 8/7 coefficient for all laws, including those with an atom at zero.

Load-bearing premise

The 8/7 coefficient rests on the point-plane incidence bound holding for arbitrary finite point and plane sets in real three-dimensional space with an absolute constant, in the unbalanced regime |G1| ≤ |G2||G3|; no explicit admissible value of that constant appears in the literature, so the stated additive constants are not numerically pinned.

What would settle it

Run a search over finite sets G1,G2,G3 ⊂ R satisfying |G1| ≤ |G2||G3| and the line condition, computing the collision count Col = #{(a,b,c,a',b',c'): a(b+c)=a'(b'+c')}; if any family has Col growing faster than a constant times (|G1||G2||G3|)^{3/2}, Theorem 5.1 and hence the 8/7 coefficient collapse. Since the constant C_R is unpinned, an explicit calculation of Col for small asymmetric sets (e.g., |G1|≈|G2||G3|) would also pin down whether the stated constants are realistic.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • This is the first proof that the entropic sum-product coefficient exceeds 1, settling the conjecture with δ = 1/7; the known upper bound is 4/3, so the optimal constant lies in [8/7, 4/3].
  • The dyadic uniformization lemma with O(log H) loss is a reusable tool: any entropy inequality currently limited by min-entropy should be re-examinable in Shannon-entropy form.
  • The corrected product-sum upper bound (Proposition 6.1) repairs an error in earlier published bounds; the paper shows the earlier main theorems survive with small modifications.
  • Uniformizing a finite set A recovers a combinatorial sum-product statement with exponent 1/7 from the entropic theorem, though the combinatorial record remains higher.
  • The small-doubling bound gives a genuinely linear entropic 'few sums force many products' inequality, stronger than the additive-constant Freiman-type bounds previously available.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same framework is likely to yield the optimal δ = 1/3 if a single linear bound with intercept 4/3 can be proved; the paper's Conjecture 1.10 states exactly the inequality that would imply it, so the bottleneck is sharpening the slope-6 coefficient in the small-doubling line.
  • Since no explicit value of the incidence constant C_R is known, the theorem's stated constants c1=18, c2=63 should be treated as schematic; the asymptotic 8/7 is unaffected, but effective versions depend on a future explicit incidence bound.
  • The order structure of the real line is used essentially in the multiplicative-energy half; the authors note the large-doubling half extends to characteristic-0 fields, suggesting a testable extension: prove an entropic sum-product bound over Q or number fields with a smaller δ, showing what loss the absence of ordering imposes.
  • Because the paper shows adding an atom at zero merely contracts the difference in Conjecture 1.10, we expect future improvements can focus on zero-free laws; the hard case is not the zero atom.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper proves the first entropic sum-product phenomenon with an explicit coefficient strictly greater than 1. Specifically, Theorem 1.5 shows that for i.i.d. discrete real-valued random variables X, X' with finite entropy, max{H(X+X'), H(XX')} ≥ (8/7)H(X) - O(log H(X)), answering Goh's Conjecture 1.2 with δ = 1/7. The proof has two main branches: a small-doubling bound (Theorem 1.6) obtained by adapting Solymosi's multiplicative-energy argument via a dyadic uniformization technique, and a large-doubling bound (Theorem 1.7) built on the Rudnev–de Zeeuw point-plane incidence bound and a corrected upper bound for H(X(Y+Z)) (Proposition 6.1). The two branches are combined in Proposition 7.2, after which an atom at zero is reincorporated and the finite-support assumption is removed by truncation. The paper also contains a detailed erratum for a flawed bound in prior work of Gavalakis–Goh–Kontoyiannis.

Significance. If correct, this is a substantial result: it gives the first coefficient strictly larger than 1 for the entropic sum-product phenomenon, resolving an open conjecture. The proof is long but essentially self-contained, and the constants are explicit up to the single absolute incidence constant C_R of Definition 2.4. The dyadic uniformization lemma (Lemma 2.7) and the linear entropic Elekes–Ruzsa theorems are likely to be useful beyond this paper. The paper also provides a careful correction of a previously published upper bound, which increases confidence in both the present proofs and the salvageability of the earlier results. The only caveat—that no explicit admissible value of C_R appears in the literature—is acknowledged by the authors and does not affect the asymptotic claim (1.3), since C_R is an absolute constant absorbed by the O(log H) term.

minor comments (4)
  1. [Definition 2.4 and Theorem 1.5] The constant C_R is defined as the implied constant in Theorem 2.3, but the paper notes that no explicit admissible value is known. Since (1.2) explicitly displays the term −(1/2)log C_R, it may be helpful to add a sentence in the introduction emphasizing that the asymptotic statement (1.3) is fully explicit while the finite-constant version is qualitative in this one additive constant.
  2. [Corollary 4.14] The extension of ϑ(η) to the range 0 < η < 4 via ϑ(η) = ϑ(4) is somewhat ad hoc. Since the main theorems are asymptotic, it might be cleaner to state the theorem for η ≥ 4 and treat small entropy separately, avoiding the artificial constant 93/4.
  3. [Section 6.3, Example 6.5] The proof of Example 6.5 is described as 'routine but tedious' and some entropy computations are summarized rather tersely, especially the claim that H(X+X' | X>0, X'>0) = log n + O(1). A few additional lines of derivation would improve readability without affecting correctness.
  4. [Lemma 7.5] The existence of a finite F with both P(X∉F) ≤ θ and Σ_{x∉F} p(x) log(1/p(x)) ≤ θ follows by convergence of the defining series, but the proof could be more explicit that one takes a sufficiently large finite initial segment after arranging the support in any order.

Circularity Check

0 steps flagged

No significant circularity: the δ = 1/7 coefficient is computed as the crossing point of two independent bounds built on external inputs (Solymosi energy, Rudnev–de Zeeuw incidence, Vadhan uniformization); self-citations are non-load-bearing.

full rationale

The central claim (Theorem 1.5) is genuinely derived, not assumed. The proof splits into: (a) a small-doubling bound em ≥ 2 − 6eu − ϑ(eH) (Corollary 4.14), built by converting Solymosi's combinatorial multiplicative-energy bound (reproduced with full proof in Theorem 3.1, quoting [17, Theorem 6] and following [25]) into an entropy statement via dyadic uniformization (Lemmas 2.6, 2.7); and (b) a large-doubling bound em ≥ 5/4 − 3/4eu − ϑ(eH) (Proposition 6.7), built from the external Rudnev–de Zeeuw incidence bound (Theorem 2.3, cited to [5]) and a corrected functional-submodularity upper bound proved in the paper (Proposition 6.1). In both branches, the target ratio em is bounded in terms of the independent additive-doubling ratio eu using only external inputs and internal bookkeeping; no equation defines the conclusion in terms of itself, and no parameter is fitted to data. The coefficient 8/7 is not selected: Proposition 7.2 proves the conclusion for the whole interval δ ≤ (Φ−1)/(Φ+1), and the computation Φ = inf F(eu)/(1−eu) = 4/3 at the intersection eu = 1/7 of the two independently derived lines F1, F2 yields δ = 1/7 purely by algebra (Theorem 7.3). Self-citation is not load-bearing: the author's prior work [19] is used only for the tightness example δ ≤ 1/3 and background, and the paper explicitly disclaims dependence on [9] ('our proofs are provided completely, i.e., do not rely on any of their results', Section 5; 'the previously stated upper bounds are actually incorrect', Section 1.1, with counterexamples in Remark 6.3). External, independently checkable inputs (Solymosi; Rudnev/de Zeeuw; Vadhan's Lemma 2.5) carry the load, and the one admitted limitation — 'No explicit admissible value of CR appears in the literature, so KR is the only unpinned absolute constant' (Definition 2.4) — affects only the additive −(1/2)log CR term in (1.2), not the asymptotic (1.3), and is a completeness caveat rather than a circular step. Per the review rule, this limitation was located and weighed, and it does not constitute circularity. No specific reduction of a conclusion to its own inputs could be exhibited; hence no circular step is reported.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The proof uses no data fitting and introduces no new physical/mathematical entities. The two listed free parameters are closed-form proof choices with no empirical content. The main external load-bearing inputs are standard entropy identities, the entropic Ruzsa triangle inequality, functional submodularity, Vadhan's uniformization lemma, and the point-plane incidence bound; the last one carries an unpinned absolute constant C_R.

free parameters (2)
  • β = 1 + 1/η (e.g., η = eH in Corollary 4.14)
    Proof-optimization parameter chosen by hand to balance the 6βe_u term against the logβ error. It is closed-form and has no empirical or data-fitting content, but the final constants depend on this choice.
  • θ = H^{-2}
    Truncation threshold chosen in Corollary 7.6 and the proofs of Theorems 1.6 and 1.7. It is chosen small enough to preserve the constants; it is a proof device, not a fitted physical parameter.
axioms (5)
  • standard math Elementary entropy facts: conditioning reduces entropy, chain rule, data processing, Gibbs inequality, support bound, max-entropy of geometric law (Lemma 2.1).
    Used throughout the paper; the geometric-law bound is proved in Lemma 2.1 and the others are standard.
  • standard math Entropic Ruzsa triangle inequality, H(X−Z) ≤ H(X−Y)+H(Y−Z)−H(Y), and its consequence (2.1).
    Cited from Ruzsa [24]; used in Lemma 4.2 to control difference entropy in terms of sum entropy and the overall doubling e_u.
  • standard math Functional submodularity: if W=f(R)=g(S) and T=φ(R,S), then H(W)+H(T) ≤ H(R)+H(S) (Lemma 2.2).
    Used in the proof of the corrected product-sum upper bound, Proposition 6.1, via data processing for mutual information.
  • domain assumption Vadhan's uniformization lemma: a law with min-entropy at least m is an exact mixture of uniform laws on sets of size 2^m (Lemma 2.5).
    Imported from Vadhan [29]; central to converting the dyadic decomposition into exact uniform pieces in Lemma 2.7 and Theorem 5.10.
  • domain assumption Rudnev–de Zeeuw point-plane incidence bound with absolute constant C_R (Theorem 2.3 and Definition 2.4).
    Central to Theorem 5.1 and hence the large-doubling bound. The paper notes no explicit admissible value of C_R appears in the literature.

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Let $X,X'$ be independent and identically distributed discrete real-valued random variables of finite Shannon entropy, and write $H(X)$ for the Shannon entropy of $X$. We prove that \[ \max\{H(X+X'),\,H(XX')\} \ge \frac87 H(X)-O(\log H(X)). \] This is the entropic analog of the celebrated sum-product phenomenon, and answers a question of Goh, which simply asked for a coefficient strictly larger than 1. An example by the author, Gavalakis, and Kontoyiannis showed the coefficient cannot exceed $\frac43$. Previous work by Gavalakis, Goh, and Kontoyiannis was able to prove a result of a weaker form, which could not translate to a coefficient strictly larger than 1 because of examples where the min-entropy is significantly smaller than the Shannon entropy. By splitting the distribution of $X$ into uniform pieces, which costs $O(\log H(X))$ entropy, we obviate this issue, establishing a coefficient of $\frac{10}{9}$. We augment this to $\frac87$ by adapting the work of Solymosi, which established the combinatorial sum-product phenomenon with coefficient $\frac43$ by bounding the multiplicative energy, to the entropy setting, again via a uniformization technique.

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