Number of solutions in a box of a linear homogeneous equation in an Abelian group. (Q2919683)

Generalizing a conjecture of \textit{A. Schinzel} [Developments in Mathematics 16, 363-370 (2008; Zbl 1239.11039)], which was proved by \textit{K. Cwalina} and \textit{T. Schoen} [Acta Arith. 153, No. 3, 271-279 (2012; Zbl 1268.11050)], the author shows the following result:NEWLINENEWLINE Let \(\Gamma\) be a finite Abelian group, \(a_1,\ldots,a_k\in\Gamma\) and \(b_1,\ldots,b_k\in\mathbb N\). Then the number of solutions \(x_i\in\mathbb Z\) with \(0\leq x_i\leq b_i\) of the equation NEWLINE\[NEWLINE\sum_{i=1}^k a_ix_i=0NEWLINE\]NEWLINE is at least NEWLINE\[NEWLINE2^{1-D(\Gamma)}\prod_{i=1}^k(b_i+1),NEWLINE\]NEWLINE where \(D(\Gamma)\) denotes the Davenport constant of \(\Gamma\). The proof is short and uses some clever identities in the group ring \(\mathbb Q[\Gamma]\).