Additive Combinatorics

Sumsets, arithmetic progressions, and the structure of approximate groups.

foundation tier

Additive Combinatorics. Sumsets, arithmetic progressions, and the structure of approximate groups.

Foundations and canonical references

The standard treatments of additive combinatorics approach the subject from complementary angles. Tao, Additive Combinatorics (2006) is the anchor reference for the subject and lays out the core definitions, theorems, and worked examples that practitioners return to.

Open methodological questions for additive combinatorics include sharpening the bridges between foundational theory and computational practice, extending classical results to broader or more structured settings, and integrating the techniques surveyed above with adjacent mathematical disciplines. The references listed in this page are the entry points that current work builds on.

Prerequisites

Sources

  • textbook · primary · 2006
    Additive Combinatorics
    tao-2006b, vu-2006

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Explore

  1. 01

    Sumset Estimates and Plünnecke–Ruzsa

    Doubling constants, Plünnecke–Ruzsa inequalities, and Freiman's theorem.
  2. 02

    Arithmetic Progressions in Sets

    Roth, Szemerédi, Green–Tao, and density results.
  3. 03

    Gowers Uniformity Norms

    Higher-order Fourier analysis and inverse theorems.

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