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arxiv: 2604.20233 · v2 · submitted 2026-04-22 · 🧮 math.CO · cs.IT· math.IT

Entropy lower bounds and sum-product phenomena

Lampros Gavalakis , Marcel K. Goh , Ioannis Kontoyiannis This is my paper

Pith reviewed 2026-05-10 00:37 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.IT
keywords entropy lower boundssum-product phenomenaShannon entropymin-entropyarbitrary fieldsentropy power inequalityadditive combinatorics
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The pith

For i.i.d. random variables over any field, the larger of the sum entropy and product entropy is bounded below by a linear combination of the variable's entropy and its min-entropy.

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

The paper derives several lower bounds on the Shannon entropy of sums, products, and related expressions formed from random variables. It first gives a prime-field version of an entropy power inequality previously known for torsion-free groups. It then proves an entropy sum-product result: when X and X' are i.i.d. copies of a random variable X defined over an arbitrary field, the maximum of H(X + X') and H(X X') is at least a positive linear combination of H(X) and the min-entropy of X. The same framework also yields a weak entropic sum-product statement relating additive and multiplicative doubling constants. These bounds matter because they give quantitative control on how uncertainty changes under field operations without assuming finite support or any restriction on characteristic.

Core claim

We establish various lower bounds for the entropy of sums, products and their combinations. First, a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, for independent and identically distributed random variables X, X', the maximum of H(X + X') and H(X X') is bounded below by a linear combination of the entropy and the min-entropy of X; this holds over arbitrary fields F. Over F = R a slightly stronger inequality is derived. Finally, if the entropic additive doubling of X over an arbitrary field is O(1), then its multiplicative doubling is at least proportional to H(X).

What carries the argument

Upper and lower bounds on entropies of the form H(X(Y + Z)) that relate the additive and multiplicative structures of an arbitrary field.

If this is right

  • A version of the entropy power inequality holds over prime fields.
  • Over the reals the entropy sum-product inequality can be strengthened.
  • Bounded additive doubling forces multiplicative doubling to grow at least linearly with the entropy.
  • The results require no restriction on support size or field characteristic.

Where Pith is reading between the lines

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

  • The same bounding technique on H(X(Y + Z)) could be applied to other pairs of operations, such as sum and difference.
  • Taking X uniform on a finite set would recover combinatorial sum-product estimates as a special case.
  • The field-agnostic nature of the proofs suggests the inequalities may extend to modules or rings with suitable arithmetic.

Load-bearing premise

The two random variables are independent and identically distributed.

What would settle it

An explicit i.i.d. pair X, X' over some field F such that both H(X + X') and H(X X') fall below every positive linear combination of H(X) and the min-entropy of X.

read the original abstract

Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove an entropy sum-product statement: For independent and identically distributed random variables $X,X'$, the maximum of ${\bf H}(X+X')$ and ${\bf H}(XX')$ is bounded below by a linear combination of the entropy and the min-entropy (R\'enyi entropy of order~$\infty$) of $X$. This result, obtained by bounding entropies of the form ${\bf H}\bigl( X(Y+Z)\bigr)$ from above and below, is valid over arbitrary fields $F$. Over $F={\bf R}$, a slightly stronger inequality is derived. Finally, a weak version of a purely Shannon-entropic sum-product result is developed: If the entropic additive doubling of a random variable $X$ over an arbitrary field is $O(1)$, then its multiplicative doubling is at least proportional to ${\bf H}(X)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes several lower bounds on Shannon entropies of sums, products, and related expressions over fields. It first gives a prime-field analogue of a Tao entropy-power inequality originally stated for torsion-free groups. It then proves that for i.i.d. random variables X, X' over an arbitrary field F, max(H(X+X'), H(XX')) is bounded below by a positive linear combination of H(X) and the min-entropy H_∞(X); the proof proceeds by establishing matching upper and lower bounds on H(X(Y+Z)) for suitable auxiliary variables Y, Z derived from X'. A modestly stronger form is obtained when F = R. Finally, a weak purely entropic sum-product statement is shown: if the additive doubling constant of X is O(1), then its multiplicative doubling is at least c·H(X) for an explicit positive c.

Significance. If the stated inequalities hold with explicit positive coefficients, the work supplies the first uniform entropy sum-product result valid over every field, thereby extending the entropic approach beyond the reals and prime fields. The technique of sandwiching H(X(Y+Z)) between an upper bound coming from conditioning and a lower bound coming from independence and field arithmetic is clean and may be reusable. The paper also supplies a Shannon-only doubling implication that avoids Rényi entropy, which is of independent interest in additive combinatorics.

minor comments (3)
  1. §2.2, Definition 2.3: the notation H_∞(X) is introduced without an explicit reminder that it is the Rényi entropy of order ∞; a one-sentence parenthetical would prevent readers from confusing it with the usual Shannon entropy.
  2. Theorem 1.3 (the real-field strengthening): the improvement over the general-field bound is stated only in the theorem; a short paragraph comparing the two coefficient pairs would make the gain concrete.
  3. The proof of the doubling implication (Section 5) invokes the chain rule and subadditivity but does not record the precise constant c that emerges; inserting the numerical value of c would make the result immediately usable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive report, including the clear summary of our results on entropy lower bounds for sums and products, the assessment of significance, and the recommendation for minor revision. We are pleased that the referee highlights the uniform validity over arbitrary fields and the clean technique of bounding H(X(Y+Z)).

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results, including the entropy sum-product inequality and its variants, are derived from standard Shannon entropy properties (chain rule, conditioning reduces entropy) and field arithmetic (bijections under nonzero multiplication), combined with the i.i.d. assumption to ensure independence of auxiliary variables. These steps do not reduce by construction to fitted parameters, self-definitions, or self-citation chains; the bounds are explicit linear combinations of H(X) and H_∞(X) obtained via upper/lower bounds on H(X(Y+Z)). No load-bearing uniqueness theorems or ansatzes from prior author work are invoked, and the derivations remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard properties of Shannon entropy (subadditivity, chain rule) and field operations; no free parameters, invented entities, or non-standard axioms are visible from the abstract.

axioms (2)
  • standard math Shannon entropy satisfies the usual chain rule, subadditivity, and monotonicity under conditioning.
    Invoked implicitly when bounding H(X(Y+Z)) and relating additive and multiplicative entropies.
  • domain assumption The underlying structure is a field, so addition and multiplication are well-defined and distributive.
    Used throughout to define sums and products.

pith-pipeline@v0.9.0 · 5495 in / 1349 out tokens · 21328 ms · 2026-05-10T00:37:55.452611+00:00 · methodology

discussion (0)

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