On semibounded expansions of ordered groups

We explore “semibounded” expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal{R}=⟨R,<,+,\dots ⟩$ is a semibounded o-minimal structure and $P\subseteq R$ a set satisfying certain tameness conditions, then $\langle \cal R, P\rangle$remainssemibounded\mathrm{.}Examplesincludethecaseswhen$$\mathcal{R}$=\langle \mathbb R,<,+, (x\mapsto \lambda x){\lambda \in \mathbb R}, \cdot{ [0, 1]^2} \rangle$,and$P= 2^\mathbb Z$or$P$ is an iteration sequence. As an application, we obtain that smooth functions definable in such $⟨\mathcal{R},P⟩$ are definable in $\mathcal{R}$.