2011/05/31 by Jörg Μ. Thuswaldner, Daniel Krenn, Jörg Thuswaldner +1 · 7 citations
Mathematics · #Algebraic Geometry and Number Theory #Algebraic number #Algebraic number field #Analytic Number Theory Research #Base (topology) #Bounded function #Combinatorics #Discrete mathematics #Field (mathematics) #Integer (computer science) #Mathematical analysis #Mathematical and Theoretical Analysis #Mathematics #Order (exchange) #Pairwise comparison #Pure mathematics #Statistics #Unit (ring theory) #math.NT #msc:11A63 #msc:11R11 #msc:11R16 #msc:11R67
paper · pdf · doi:10.1112/s0025579311001884
published in Mathematika 57(2), 247-262 (Wiley)
openalex publication_date 2011/05/31 · arxiv created 2012/07/24 · arxiv updated 2014/01/14 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/29
Starting with a paper of Jacobson from the 1960s, many authors became interested in characterizing all algebraic number fields in which each integer is the sum of pairwise distinct units. Although there exist many partial results for number fields of low degree, a full characterization of these number fields is still not available. Narkiewicz and Jarden posed an analogous question for sums of units that are not necessarily distinct. In this paper we propose a generalization of these problems. In particular, for a given rational integer n we consider the following problem. Characterize all number fields for which every integer is a linear combination of finitely many units εi in a way that the coefficients ai∈ℕ are bounded by n. The paper gives several partial results on this problem. In our proofs we exploit the fact that these representations are related to symmetric beta expansions with respect to Pisot bases.