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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.

##### MSC:
 11B25 Arithmetic progressions 20K01 Finite abelian groups 11B75 Other combinatorial number theory
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##### References:
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