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On subsets of finite Abelian groups with no 3-term arithmetic progressions. (English) Zbl 0832.11006
For a finite commutative group \(G\) let \(D(G)\) denote the maximal cardinality of subsets \(A\subset G\) that do not contain a 3-term arithmetic progression. Heath-Brown and Szemerédi’s improvement of Roth’s theorem can be formulated as \(D(\mathbb{Z}_m) \ll m(\log m)^{-c}\). Here this is extended to all groups in the form \(D(G) \ll|G|(\log |G|)^{-c}\). The key ingredient is the following result. If \(G= \mathbb{Z}_{k_1} \otimes \dots \otimes \mathbb{Z}_{k_n}\) with \(k_1 |\dots|k_n\), then \(D(G)\leq 2|G|/n\). Thus, for instance, for \(G= \mathbb{Z}^n_3\) the exponent is \(c=1\). (On the other hand, no analog of Behrend’s construction is known for such groups, hence it is possible that the real order is \(D(G) \ll|G|^{1- \delta}\) with some \(\delta >0\).) The proof uses a discrete version of Roth’s method.

11B25 Arithmetic progressions
20K01 Finite abelian groups
11B75 Other combinatorial number theory
Full Text: DOI
[1] Alon, N; Dubiner, M, Zero-sum sets of prescribed size, (), 33-50 · Zbl 0823.11006
[2] {\scN. Alon and M. Dubiner}, A lattice point problem and additive number theory, to appear. · Zbl 0838.11020
[3] Brown, T.C; Buhler, J.C, A density version of a geometric Ramsey theorem, J. combin. theory ser. A, 32, 20-34, (1982) · Zbl 0476.51008
[4] Frankl, P; Graham, R; Rödl, V, On subsets of abelian groups with no 3-term arithmetic progression, J. combin. theory ser. A, 45, 157-161, (1987) · Zbl 0613.10043
[5] Heath-Brown, R, Integer sets containing no arithmetic progressions, J. London math. soc., 35, 385-394, (1987) · Zbl 0589.10062
[6] Roth, K.F, On certain sets of integers, J. London math. soc., 28, 104-109, (1953) · Zbl 0050.04002
[7] Szemerédi, E, Integer sets containing no arithmetic progressions, Acta math. hungar., 56, 155-158, (1990) · Zbl 0721.11007
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