Arithmetics of beta-expansions
Louis-Sébastien Guimond, Zuzana Masáková, Edita Pelantová · Acta Arithmetica · 2004
In this paper we consider representation of numbers in an irrational basis β> 1. We study the arithmetic operations on β-expansions and provide bounds on the number of fractional digits arising in addition and multiplication, L⊕(β) and L(β), respectively. We determine these bounds for irrational numbers β which are algebraic with at least one conjugate in modulus smaller than 1. In the case of a Pisot number β we derive the relation between β-integers and cut-and-project sequences and then use the properties of cut-and-project sequences to estimate L⊕(β) and L(β). We generalize the results known for quadratic Pisot units to other quadratic Pisot numbers. 1 Beta-expansions Let β be a real number strictly greater than 1. A real number x ≥ 0 can be represented using a sequence (xi)k≥i>−∞, xi ∈ Z, 0 ≤ xi < β, such that