Annals of Mathematics· 2016Q1
On large subsets of $\mathbb{F}_q^n$ with no three-term arithmetic progression
- 141citations
- Q1SCImago
- 2016year
Short summary
A new method establishes that subsets of $\mathbb{F}_q^n$ lacking three-term arithmetic progressions can have size at most $c^n$ where $c < q$.
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Key points
- Introduces a new upper bound for subsets of $\mathbb{F}_q^n$ without three-term arithmetic progressions.
- The bound is $c^n$, where $c < q$.
- Applies the Croot-Lev-Pach method to this problem.
- Improves the prior bound for the $\mathbb{F}_3^n$ cap set problem from $n^{-1-\epsilon}$ to $c^n$ with $c < 3$.
AI-generated from the title and abstract; the full text is not read.
Abstract
In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of F n q Fqn with no three terms in arithmetic progression by c n cn with c&lt;q c&lt;q . For q=3 q=3 , the problem of finding the largest subset of F n 3 F3n with no three terms in arithmetic progression is called the <i>cap set problem</i>. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz, was on order n −1−ϵ 3 n n−1−ϵ3n .
The authors' abstract, as published at the source. Annals of Mathematics, 2016 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics