Treffer: Restricted sums in a field

Let \(A_1, \dots , A_n\) be finite sets in a field of characteristic \(p\). The authors investigate the cardinality of a generalized restricted sumset of the type \[ C = \{a_1+\dots +a_n: a_i\in A_i, a_i-a_j \not\in S_{ij} \} \] with given sets \(S_{ij}\) for \(1\leq i,j\leq n\), \(i\neq j\). If \(S_{ij}=\{0\}\) for all \(i,j\), this reduces to the sum of distinct elements. The main result asserts that \[ |C |\geq (k+m-mn-1)n+1 \tag{1} \] under the assumptions that \( |A_i |=k\) and \( |S_{ij} |\leq m\) for all \(i,j\), and either \(p=0\) or \(p\) is sufficiently large. The last condition means that \(p\) must exceed the bound in (1) and also that \(p>mn\), the relevance of which is unclear. It is not settled whether \( |C |\geq p\) holds when \(p\) is less than the bound in (1), though a remark gives some result in this case. The authors present some cases of equality of their bound and formulate a conjecture that would extend this result to the case when the cardinalities of the sets \(A_i\) are not necessarily equal. The proof applies the polynomial method as developed by \textit{N. Alon, M. B. Nathanson} and \textit{I. Z. Ruzsa} [J. Number Theory 56, 404--417 (1996; Zbl 0861.11006)]. In this generality, however, the calculation of the necessary coefficients is considerably more difficult.