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Abstract:

Let α, β ∈ ℝ_{>0} be such that α, β are quadratic and ℚ(α) ≠ ℚ(β). Then every subset of ℝ^n definable in both (ℝ,<,+,ℤ,x → α x) and (ℝ, < ,+,ℤ,x → β x) is already definable in (ℝ, < ,+,ℤ). As a consequence we generalize Cobham-Semenov theorems for sets of real numbers to β-numeration systems, where β is a quadratic irrational.

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Full text arXiv 2407.15118: pdf, ps.