Geodesic
On the structure of sets with few three-term arithmetic progressions
The electronic journal of combinatorics, Tome 17 (2010)
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Fix a prime $p \geq 3$, and a real number $0 < \alpha \leq 1$. Let $S \subset {\mathbb F}_p^n$ be any set with the least number of solutions to $x + y = 2z$ (note that this means that $x,z,y$ is an arithmetic progression), subject to the constraint that $|S| \geq \alpha p^n$. What can one say about the structure of such sets $S$? In this paper we show that they are "essentially" the union of a small number of cosets of some large-dimensional subspace of ${\mathbb F}_p^n$.
DOI : 10.37236/400
Classification : 11B25, 11B30, 11N30 
@article{10_37236_400,
     author = {Ernie Croot},
     title = {On the structure of sets with few three-term arithmetic progressions},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/400},
     zbl = {1252.11010},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/400/}
}
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Ernie Croot. On the structure of sets with few three-term arithmetic progressions. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/400

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