Defining the Integers in Expansions of the Real Field by a Closed Discrete Set
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چکیده
Let D ⊆ R be closed and discrete and f : Dn → R be such that f(Dn) is somewhere dense. We show that (R,+, ·, f) defines Z. As an application, we get that for every α, β ∈ R with log α (β) / ∈ Q, the real field expanded by the two cyclic multiplicative subgroups generated by α and β defines Z.
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Caveat. This note has become seriously out of date due, among other things, to recent work by Philipp Hieronymi [Defining the set of integers in expansions of the real field by a closed discrete set. Proc. Amer. Math. Soc. 138 (2010), no. 6, 21632168. MR2596055]. In particular, the paragraph after the proof of Corollary 4 should be struck. Rather than update this particular note, we are are act...
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تاریخ انتشار 2009