The real field with an irrational power function and a dense multiplicative subgroup

TL;DR

This paper constructs a well-behaved mathematical structure combining the real field with an irrational power function and a dense multiplicative subgroup, proving quantifier elimination and definability results under Schanuel conditions.

Contribution

It provides the first example of such a structure with these properties, extending o-minimal theory with dense subgroups and irrational powers.

Findings

Quantifier elimination under Schanuel conditions

Open sets definable in the structure are already definable in the reduct

The structure is model-theoretically well-behaved

Abstract

This paper provides a first example of a model theoretically well behaved structure consisting of a proper o-minimal expansion of the real field and a dense multiplicative subgroup of finite rank. Under certain Schanuel conditions, a quantifier elimination result will be shown for the real field with an irrational power function and a dense multiplicative subgroup of finite rank whose elements are algebraic over the field generated by the irrational power. Moreover, every open set definable in this structure is already definable in the reduct given by just the real field and the irrational power function.