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On large subsets of \(\mathbb{F}_q^n\) with no three-term arithmetic progression. (English) Zbl 1425.11020
Summary: In this note, we show that the method of E. Croot, V. F. Lev, and P. P. Pach [Ann. Math. (2) 185, No. 1, 331–337 (2017; Zbl 1425.11019)] can be used to bound the size of a subset of \(\mathbb{F}_q^n\) with no three terms in arithmetic progression by \(c^n\) with \(c<q\). For \(q=3\), the problem of finding the largest subset of \(\mathbb{F}_3^n\) with no three terms in arithmetic progression is called the cap set problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz, was on order \(n^{-1-\varepsilon} 3^n\).

MSC:
11B30 Arithmetic combinatorics; higher degree uniformity
11B25 Arithmetic progressions
51E20 Combinatorial structures in finite projective spaces
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