CNRS-Pr1: Toward a Definition of Base−b for All b∈R+: Extending the Ito–Sadahiro (−β)-Expansion Framework

We work toward a rigorous definition of base −b for all b ∈ R+, extending the (−β) -expansion framework of Ito and Sadahiro (2009). This problem — designated Problem 1 of the Complex Numeric Representational System programme 11 — is the necessary first step toward a numeric system in which complex numbers are represented as single values. We identify the precise location of the β ≤ 2 restriction in the Ito–Sadahiro construction: it lives in the digit alphabet, not in the interval width. The interval Iβ = −β/ (β + 1), 1/ (β + 1) ) has width 1 for all β > 1 and defines the correct fundamental domain. The natural generalisation for β ∈ (n, n + 1 retains this interval but extends the digit alphabet to Σ (β) = 0, 1,. . . , n, and replaces the greedy map with T−β (x) = −βx − ⌊−βx − ℓβ⌋, where ℓβ = −β/ (β + 1) is the left endpoint. We carry out complete, corrected orbit computations for two Pisot numbers in (2, 3]: β = 1 + √2 (Theorem 6. 2) and β = 1 + √3 (Theorem 7. 2). In both cases the endpoint orbits are eventually periodic and explicit, decidable admissibility conditions are given. The admissibility blocks differ between the two cases, determined by the respective minimal polynomials; the blocks are (2, 1^ω) and (0, 2, 1^ω) for β = 1 + √2, and (2, 0^ω) and (0, 2, 0^ω) for β = 1 + √3. 4 We formulate the Pisot periodicity conjecture: eventual periodicity of the endpoint orbits is equivalent to the Pisot property. We state three open sub-questions constituting the full statement of Problem 1. Changes in version 6. The interval and map definitions have been corrected following correspondence with W. Steiner (IRIF, Paris). Version incorrectly used the extended interval [−n/ (β+1), 1/ (β+1) ) and the unshifted map T−β (x) = −βx − ⌊−βx⌋. These errors caused the admissibility blocks and all associated conjectures in v5 to be incorrect. Theorems, orbit tables, and conjectures have been completely restated. The invariance gap and exit strip analysis of v5 (Section 8 of that document) have been removed, as they were artefacts of the incorrect interval.