Finite field models in additive combinatorics

By Ben Green. 2004

76 citations recorded by the source corpus.

The study of many problems in additive combinatorics, such as Szemer\'edi's theorem on arithmetic progressions, is made easier by first studying models for the problem in F_p^n for some fixed small prime p. We give a number of examples of finite field models of this type, which allows us to introduce some of the central ideas in additive combinatorics relatively cleanly. We also give an indication of how the intuition gained from the study of finite field models can be helpful for addressing the original questions.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/006149e0eb35e7b7a187dc4073cfb6ce0ebb1cec

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Terence Tao
2. Tom Sanders
3. William T. Gowers
4. Ben Joseph Green
5. Julia Wolf
6. Alfred Geroldinger
7. Christian Elsholtz
8. Wolfgang A. Schmid

In Discrete Mathematics and Combinatorics

1. Terence Tao
2. William T. Gowers
3. Ben Joseph Green
4. Tom Sanders
5. Julia Wolf
6. Nikos Frantzikinakis
7. B. Host
8. T. Ziegler

In Geometry and Topology

1. Terence Tao
2. Tom Sanders
3. Ben Joseph Green
4. William T. Gowers
5. Julia Wolf
6. Nikos Frantzikinakis
7. Alfred Geroldinger
8. B. Host

Citations in the corpus

• Zero-sum problems in finite abelian groups and affine caps (2006, background): “Green [33] shows all four proofs can be adapted to give an upper bound of O 3r r for the cardinality of maximal sets in F3 without an arithmetic progression of length 3.”
• A quantitative version of the idempotent theorem in harmonic analysis (2006, methodology): “This phenomenon is discussed in detail in the survey [7].”
• The true complexity of a system of linear equations (2007, background): “However, in the quadratic case, this problem has been remedied by Green and Tao [9].”
• Higher Order Fourier Analysis (2012, background): “6] or [Gr2005a] for further discussion of this point.”
• New bounds for Szemeredi's theorem, II: A new bound for r_4(N) (2006, methodology): “We have mentioned, albeit briefly, the so-called finite field model: the survey [7] may be consulted for more information.”
• On a conjecture of Marton (2023, unknown): “We refer to [7, 8, 20] for surveys on this conjecture.”
• COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN–BILU THEOREM (2005, unknown): “It has only been properly formulated in the finite field setting [10], and indeed some care is needed when working in Z.”