An addition theorem and its arithmetical application
type: Item
instance of: scholarly article
title: An addition theorem and its arithmetical application
zbMATH Open document ID: 1019.11002
author: Alfred Geroldinger
author: G. A. Freiman
published in: Journal of Number Theory
publication date: 2001-01-01T00:00:00Z
review text: 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 \(M^*\).
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 1547373
zbMATH Keywords: almost arithmetical multiprogression
zbMATH Keywords: lengths of factorizations of algebraic integers
zbMATH Keywords: finite sumset
reviewed by: Franz Halter-Koch
MaRDI profile type: Publication
full work available at URL: jnth.2000.2521
OpenAlex ID: W2038825478
cites work: Factorization in Integral Domains, IV
cites work: On non-unique factorizations into irreducible elements
cites work: Chains of factorizations and sets of lengths
cites work: A structure theorem for sets of lengths
cites work: Elasticity of factorizations in atomic monoids and integral domains
cites work: Q4001587
DOI: 10.1006/JNTH.2000.2521
description: scientific article; zbMATH DE number 1547373