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Information Inequalities via Ideas from Additive Combinatorics

Chin Wa Lau, Chandra Nair

TL;DR

This paper develops a formal bridge between sumset inequalities in additive combinatorics and entropic inequalities in information theory by generalizing Ruzsa-type equivalence to finitely generated torsion-free abelian groups. It introduces a generalized equivalence theorem linking combinatorial and entropic inequalities (and their variants), a Katz–Tao sum-difference information framework, and an entropic characterization of the magnification ratio for graphs. A key technical contribution is an entropic copy-lemma and uniformity results that underpin the equivalence and magnification analyses, with implications for subadditivity questions and network information theory. Overall, the work provides a toolkit to translate between combinatorial structure and information-theoretic constraints, enabling new entropic proofs and potential generalizations in both fields.

Abstract

Ruzsa's equivalence theorem provided a framework for converting certain families of inequalities in additive combinatorics to entropic inequalities (which sometimes did not possess stand-alone entropic proofs). In this work, we first establish formal equivalences between some families (different from Ruzsa) of inequalities in additive combinatorics and entropic ones. As a first step to further these equivalences, we establish an information-theoretic characterization of the magnification ratio that could also be of independent interest.

Information Inequalities via Ideas from Additive Combinatorics

TL;DR

This paper develops a formal bridge between sumset inequalities in additive combinatorics and entropic inequalities in information theory by generalizing Ruzsa-type equivalence to finitely generated torsion-free abelian groups. It introduces a generalized equivalence theorem linking combinatorial and entropic inequalities (and their variants), a Katz–Tao sum-difference information framework, and an entropic characterization of the magnification ratio for graphs. A key technical contribution is an entropic copy-lemma and uniformity results that underpin the equivalence and magnification analyses, with implications for subadditivity questions and network information theory. Overall, the work provides a toolkit to translate between combinatorial structure and information-theoretic constraints, enabling new entropic proofs and potential generalizations in both fields.

Abstract

Ruzsa's equivalence theorem provided a framework for converting certain families of inequalities in additive combinatorics to entropic inequalities (which sometimes did not possess stand-alone entropic proofs). In this work, we first establish formal equivalences between some families (different from Ruzsa) of inequalities in additive combinatorics and entropic ones. As a first step to further these equivalences, we establish an information-theoretic characterization of the magnification ratio that could also be of independent interest.
Paper Structure (13 sections, 34 theorems, 139 equations)

This paper contains 13 sections, 34 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Main
  5. Equivalence between sumset inequalities and entropic inequalities
  6. Katz-Tao Sum Difference Inequality
  7. Entropic Formulation of Magnification Ratio
  8. Proof of \ref{['lem:ent-uniform']}
  9. Conclusion
  10. Proof of Theorem \ref{['th:katz-tao-ent']}
  11. Discrete Sanov Theorem
  12. Types
  13. Maximal couplings

Key Result

Theorem 1

(Generalized Ruzsa-type Equivalence Theorem) Let $(\mathbb{T},+)$ be a finitely generated torsion-free abelian group. Let $f_1, \dots, f_k$ and $g_1, \dots, g_\ell$ be linear functions on $\mathbb{T}^n$ with integer coefficients, and let $\alpha_1, \dots, \alpha_k, \beta_1, \dots, \beta_\ell$ be pos

Theorems & Definitions (86)

  • Theorem 1
  • proof
  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Corollary 1
  • proof
  • Remark 3
  • Corollary 2
  • ...and 76 more