We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885-903]. Let G be an abelian semialgebraic group over a real closed field R and let X be a semialgebraic subset of G. Then the group generated by X contains a generic set and, if connected, it is divisible. More generally, the same result holds when X is definable in any o-minimal expansion of R which is elementarily equivalent to a an,exp. We observe that the above statement is equivalent to saying: there exists an m such that ςi=1m(X - X) is an approximate subgroup of G.
Bibliographical noteFunding Information:
We thank Eliana Barriga for reading and commenting on an early version of this paper. We also thank the anonymous referee for suggestions which improved the presentation of this paper. E. Baro was supported by the program MTM2017-82105-P. P. E. Eleftheriou was supported by an Independent Research Grant from the German Research Foundation (DFG) and a Zukunftskolleg Research Fellowship.
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- Semialgebraic groups
- approximate groups
- generic sets
- locally definable groups
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