Defining additive subgroups of the reals from convex subsets

Tychonievich, Michael

doi:10.1090/S0002-9939-09-09914-6

Defining additive subgroups of the reals from convex subsets
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by Michael A. Tychonievich PDF

Proc. Amer. Math. Soc. 137 (2009), 3473-3476 Request permission

Abstract:

Let $G$ be a subgroup of the additive group of real numbers and let $C\subseteq G$ be infinite and convex in $G$. We show that $G$ is definable in $(\mathbb R,+,\cdot ,C)$ and that ${\mathbb Z}$ is definable if $G$ has finite rank. This has a number of consequences for expansions of certain o-minimal structures on the real field by multiplicative groups of complex numbers.

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