Chromatic sumsets (Q2212621)

This paper is a contribution to additive combinatorics. More precisely the main contribution is an inverse result for sumsets. Write \(\mathbf{A} = (A_1, \ldots, A_q)\) for a \(q\)-tuple of finite sets of integers. To each such \(\mathbf{A}\), associate a \(q\)-tuple of nonnegative integers \(\mathbf{h} (h_1, \dots, h_q)\). Finally, define the set \[\mathbf{h}\cdot \mathbf{A} = h_1A_1 + \cdots + h_qA_q,\] where \(h_i A_i\) denotes the dilate set. Consider now the set \((\textbf{h}\cdot \textbf{A})^{(t)}\) consisting of all elements in \(\mathbf{h}\cdot \mathbf{A}\) with at least \(t\) representations. The main result of the paper is a structural result for this set for all sufficiently large \(h_i\). The main result on the paper (Theorem 2) generalizes previous works by \textit{S.-P. Han} et al. [Ramanujan J. 2, No. 1--2, 271--281 (1998; Zbl 0911.11008)].