Defining additive subgroups of the reals from convex subsets

Abstract

Let G be a subgroup of the additive group of real numbers and let C ⊆ G be infinite and convex in G. We show that G is definable in (R, +,.,C) and that 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.