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

Abstract

In this paper we study the expansions of real numbers in positive and negative real base as introduced by Rényi, and Ito & Sadahiro, respectively. In particular, we compare the sets Z + β and Z −β of nonnegative β-integers and (−β)-integers. We describe all bases (±β) for which Z + β and Z −β can be coded by infinite words which are fixed points of conjugated morphisms, and consequently have the same language. Moreover, we prove that this happens precisely for β with another interesting property, namely that any integer linear combination of non-negative powers of the base −β with coefficients in {0, 1, . . . , ⌊β⌋} is a (−β)-integer, although the corresponding sequence of digits is forbidden as a (−β)-integer.

Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

“…From there we derive that the distances in X {0,1} (−τ) = τ −2 Σ τ ′2 ,τ 2 [0, 1) take values 1 and τ −1 . It is known [8] that the spectrum X {0,1} (−τ) coincides with the set Z −β of (−τ)-integers and that the infinite word over the alphabet A = {A, B}, l A = 1, l B = τ −1 coding the ordering of distances in Z −β is a fixed point of the substitution A → AAB, B → AB, with the initial pair A|B. This identifies the spectrum X {0,1} (−τ) with a geometric representation of the fixed point presented in Example 10.…”