An addition theorem and its arithmetical application
Let \(0 \leq d \leq M\) be integers and \(\{0,d\} \subset D \subset [0,d] \cap \mathbb Z\). A subset \(L \subset \mathbb Z\) is called an almost arithmetical multiprogression (AAMP) with bound \(M\) and pattern \(D\), if \(y + L = L' \cup L^* \cup L'' \subset D + d \mathbb Z\), where \(y \in \mathbb Z, \;L' \subset [-M,0), \;L'' \subset \max L^* + (0,M]\) and \(0 \in L^* \subset [0,B] \cap (D+d \mathbb Z)\) for some \(B \in \mathbb N\). AAMPs appear as sets of lengths of factorizations of algebraic integers [see \textit{D. D. Anderson} (ed.), Factorization in integral domains, Lect. Notes Pure Appl. Math. 189, M. Dekker (1997; Zbl 0865.00039)]. Let \(\mathcal L\) be any set of AAMPs with the same bound \(M\). Then there exists some \(M^* \in \mathbb N\) such that every finite sumset \(L_1 + \ldots + L_n\) with \(L_i \in \mathcal L\) is an AAMP with bound \(M^*\).
lengths of factorizations of algebraic integers
scientific article; zbMATH DE number 1547373