2010/02/04 by Zuzana Masáková, Edita Pelantová, Masáková, Z. +3 · 1 citation
Computer Science · Mathematics · #11A63 #11K16 #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1002.1009
openalex publication_date 2010/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set \rm Fin(-β) and \Z-β of numbers having finite resp. integer (-β)-expansions. We show that \rm Fin(-β) is trivial if β is smaller than the golden ratio \frac12(1+√5). For β≥\frac12(1+√5) we prove that \rm Fin(-β) is a ring, only if β is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that \rm Fin(-β) is a ring if β is a quadratic Pisot number with positive conjugate. For quadratic Pisot units we determine the number of fractional digits that may appear when adding or multiplying two (-β)-integers.