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 x 2 − mx − n, m ≥ n ≥ 1, and show that in this case the set Fin(−β) of finite (−β)-expansions is closed under addition, although it is not closed under subtraction. A particular example is β = τ = 1 2 (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 (τ 2

We study periodic expansions in positional number systems with a base β ∈ C, |β| > 1, and with coefficients in a finite set of digits A ⊂ C. We are interested in determining those algebraic bases for which there exists A ⊂ Q(β), such that all elements of Q(β) admit at least one eventually periodic representation with digits in A. In this paper we prove a general result that guarantees the existence of such an A. This result implies the existence of such an A when β is a rational number or an algebraic integer with no conjugates of modulus 1. We also consider periodic representations of elements of Q(β) for which the maximal power of the representation is proportional to the absolute value of the represented number, up to some universal constant. We prove that if every element of Q(β) admits such a representation then β must be a Pisot number or a Salem number. This result generalises a well known result of Schmidt (Bull Lond Math Soc 12(4): [269][270][271][272][273][274][275][276][277][278] 1980).

No abstract

No abstract

We consider numeration systems with base β and −β, for quadratic Pisot numbers β and focus on comparing the combinatorial structure of the sets Z β and Z −β of numbers with integer expansion in base β, resp. −β. Our main result is the comparison of languages of infinite words u β and u −β coding the ordering of distances between consecutive β-and (−β)-integers. It turns out that for a class of roots β of x 2 − mx − m, the languages coincide, while for other quadratic Pisot numbers the language of u β can be identified only with the language of a morphic image of u −β . We also study the group structure of (−β)-integers.