2010/11/05 by Z. Masáková, Masáková, Z., T. Vávra +1
Computer Science · Mathematics · #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #cs.DM #math.NT
paper · pdf · doi:10.48550/arxiv.1011.1403
arxiv created 2010/11/05 · arxiv updated 2010/11/08
We consider positional numeration system with negative base -β, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when β is a quadratic Pisot number. We study a class of roots β>1 of polynomials x2-mx-n, m≥ n≥ 1, and show that in this case the set \rm Fin(-β) of finite (-β)-expansions is closed under addition, although it is not closed under subtraction. A particular example is β=τ=\frac12(1+√5), the golden ratio. For such β, we determine the exact bound on the number of fractional digits appearing in arithmetical operations. We also show that the set of (-τ)-integers coincides on the positive half-line with the set of (τ2)-integers.