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=\langle \mathbb R, \lt , +, \ldots \rangle $ is a semibounded o-minimal structure and $P\subseteq \mathbb R$ is a set satisfying certain tameness conditions, then $\langle \mathcal R, P\rangle $ remains semibounded. Examples include the cases when $\mathcal R=\langle \mathbb R, \lt ,+, (x\mapsto \lambda x)_{\lambda \in \mathbb R}, \cdot _{\upharpoonright [0, 1]^2}\rangle $, and $P= 2^\mathbb Z$ or $P$ is an iteration sequence. As an application, we show that smooth functions definable in such $\langle \mathcal R, P\rangle $ are definable in $\mathcal R$.