Arithmetics of beta-expansions

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 1. We denote L⊕(β) := min{L ∈ N0 | ∀x, y ∈ Zβ , x+ y ∈ Fin(β) =⇒ fpβ(x+ y) ≤ L} , L (β) := min{L ∈ N0 | ∀x, y ∈ Zβ , xy ∈ Fin(β) =⇒ fpβ(xy) ≤ L} . Minimum of an empty set is defined to be +∞. The aim of this paper is to give some quantitative results for L⊕(β) and L (β). Let us mention some of the known results. Frougny and Solomyak in [4] showed that L⊕(β) is finite if β is a Pisot number. A Pisot number β is an algebraic integer such that β > 1 and all its algebraic conjugates are in modulus smaller than 1. Let us mention that according to the knowledge of authors no example is known of such β that L⊕(β) or L (β) is infinite. Results for special case of quadratic Pisot units are found in [3]. The authors gave exact values for L⊕(β) and L (β), if β > 1 is a root of equation x = mx− 1, m ∈ N, m ≥ 3 or equation x = mx+ 1, m ∈ N. In the first case L⊕(β) = L (β) = 1; in the second case L⊕(β) = L (β) = 2. In this article we provide estimates on L⊕(β), L (β) for those algebraic numbers β > 1 that have at least one of the conjugates in modulus smaller than 1. Other results are valid for Pisot numbers β. The last part of the paper is devoted to quadratic Pisot numbers. We reproduce the results of [3] as a special case. 2 Beta-integers and cut-and-project sequences The Renyi development of unity plays an important role in the description of properties of sets Zβ and Fin(β). For its definition we introduce the transformation Tβ(x) := {βx}, for x ∈ [0, 1]. The Renyi development of unity is defined as d(1, β) := t1t2 . ti . , where ti := [βT i−1 β (1)] . Parry in [6] has showed that x = xkxk−1 . x1x0 • x−1 . x−p is a β-expansion if and only if xixi−1 . x−p is lexicographically smaller than t1t2 . ti . for every −p ≤ i ≤ k. Fin(β) and Zβ are centrally symmetric sets. While Fin(β) is dense in R, Zβ has no accummulation points. Distances between consecutive points in Zβ take values {0 • titi+1 . | i ∈ N}. It is obvious that if d(1, β) is eventually periodic, then Zβ has a finite number of distances between consecutive points. Numbers β with this property are called beta-numbers. Some results and conjectures on beta-numbers are given in [2, 9]; description of beta-numbers is provided in [8]. Note that every Pisot number β is a beta-number. The set Zβ of β-integers forms a ring only in the case that β is a rational integer, β > 1. If β is an algebraic integer of order q ≥ 2, then Zβ can be naturally embedded into the ring Z[β] defined as Z[β] := {n0 + n1β + · · ·+ nq−1β | ni ∈ Z} .