Restricted sums of subsets of \({\mathbb Z}\) (Q2717765)

Let \(A_1,\dots , A_n\) be finite sets of integers, and let \(V\) be a set of quintuples of integers. The author considers the \(V\)-restricted sumset \(C\), defined as the set of all sums \(a_1+\dots +a_n\) where \(a_i\in A_i\) and it is assumed that \(\mu a_i+\nu a_j\neq w\) for each \((i,j,\mu ,\nu ,w)\in V\). The main result of the paper asserts that NEWLINE\[NEWLINE |C |\geq \sum |A_i |-2 |V |-n+1 NEWLINE\]NEWLINE under the natural restriction that NEWLINE\[NEWLINE |\{ (s,t,\mu ,\nu ,w)\in V: i\in \{s,t\}\} |< |A_i |NEWLINE\]NEWLINE for each \(i\). The cases of equality are described in some interesting subcases.