Submission date: 27 December 2018

Abstract:

We prove the following instance of a conjecture stated in arXiv:1103.4770 (Preprint 312 of MODNET server).

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 in particular is divisible. More generally, the same result holds when X is definable in any o-minimal expansion of R which is elementarily equivalent to the real exponential field with restricted analytic functions. We observe that the above statement is equivalent to saying: there exists an m such that the sum of the set X-X to itself m times is an approximate subgroup of G.

Mathematics Subject Classification: 03C64, 03C68, 22B99, 20N99

Keywords and phrases: semialgebraic groups, locally definable groups, approximate groups, generic sets, lattices

Full text arXiv 1812.10682: pdf, ps.