On subsets of finite Abelian groups with no 3-term arithmetic progressions
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.
scientific article; zbMATH DE number 774653