A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression. II
Let \(G\simeq {\mathbb Z}/k_1 {\mathbb Z}\oplus\ldots \oplus{\mathbb Z}/k_N {\mathbb Z}\) be a finite abelian group with \(k_i\mid k_{i-1},2\leq i\leq N\). For a matrix \(Y=(a_{ij})\in {\mathbb Z}^{R\times S} \) satisfying \[ a_{i,1}+\ldots+a_{i,S}=0, 1\leq i\leq R \] let \(D_Y(G)\) denote the maximal cardinality of a set \(A\subseteq G\) for which the equations \[ a_{i,1}x_1+\ldots+ a_{i,S}x_S=0, 1\leq i\leq R \] are never satisfied by distinct elements \(x_1,\ldots,x_S\in A\). Under certain assumptions on \(Y\) and \(G\), the authors prove that \[ D_Y(G)\leq |G|\left(\frac{C}{N}\right)^\gamma \] for positive constants \(C\) and \(\gamma\). For the first part see [\textit{Y.-R. Liu, C. V. Spencer}, Des. Codes Cryptography 52, No. 1, 83--91 (2009; Zbl 1222.11013)].
scientific article; zbMATH DE number 5833959