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The Lens of Abelian Embeddings

Dor Minzer

Abstract

We discuss a recent line of research investigating inverse theorems with respect to general k-wise correlations, and explain how such correlations arise in different contexts in mathematics. We outline some of the results that were established and their applications in discrete mathematics and theoretical computer science. We also mention some open problems for future research.

The Lens of Abelian Embeddings

Abstract

We discuss a recent line of research investigating inverse theorems with respect to general k-wise correlations, and explain how such correlations arise in different contexts in mathematics. We outline some of the results that were established and their applications in discrete mathematics and theoretical computer science. We also mention some open problems for future research.
Paper Structure (36 sections, 18 theorems, 27 equations)

This paper contains 36 sections, 18 theorems, 27 equations.

Table of Contents

  1. Introduction.
  2. Arithmetic Progressions.
  3. Arithmetic Progressions of Length $3$.
  4. Longer Progressions.
  5. Constraint Satisfaction Problems.
  6. Decision and Optimization Problems.
  7. The Almost Satisfiable Regime.
  8. Dictatorship Tests.
  9. The Satisfiable Regime.
  10. The Main Analytical Questions.
  11. The Case of No Abelian Embeddings.
  12. The Case of Abelian Embeddings.
  13. Inverse Theorems for $k$-Wise Correlations: Statements.
  14. The Case of $3$-ary Distributions with No Abelian Embeddings.
  15. The Case of $3$-ary Distributions with Abelian Embeddings.
  16. ...and 21 more sections

Key Result

Theorem 1.4

For any prime $p$ and $\varepsilon>0$, the problem gap-$3$-Lin$[1-\varepsilon,1/p+\varepsilon]$ is NP-hard.

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Hå stad Hastad
  • Theorem 1.5: Raghavendra
  • Definition 2.2
  • Definition 2.3
  • Definition 2.5
  • Theorem 3.1
  • Theorem 3.2
  • ...and 22 more