Integers in number systems with positive and negative quadratic Pisot base

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.

Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems

“…Proof. We can assume the existence of at least three distinct distances in Z + β and Z −β , since the case with one distance corresponds to integer bases and two distances correspond to quadratic bases, already solved in [21].…”

“…Proof. Thanks to Proposition 14, relation (17), and results in [21], we can consider only simple Parry numbers β > 1 with…”

“…We already know from Example 9 that the sets of distances in Z + β and Z −β do not coincide, hence ϕ 2 ≁ ψ 2 . However, one can use similar approach as in [21] to show that cutting every distance…”

“…Quadratic and cubic numbers. The comparison of (±β)-integers for quadratic numbers β was done in [21]. It was shown that the sets of distances in Z + β and in Z −β coincide if and only if ϕ 2 ∼ ψ 2 , which is equivalent to Z −β = X(−β).…”

“…, ⌊β⌋} is a (−β)-integer? Such a question was already put forth in [21], where it is shown that among quadratic numbers, only zeros of x 2 − mx − m, m ≥ 1, have the required property. Moreover, these are exactly the bases for which the distances between consecutive points in Z β and Z −β take the same values ≤ 1.…”