Defining new linear functions in tame expansions of the real ordered additive group

We explore \emph{semibounded} expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a \emph{semibounded} expansion of an arbitrary ordered group, extending the usual notion from the o-minimal setting. For $\mathcal{R}=( \mathbb{R}, <, +, \ldots)$, a semibounded o-minimal structure and $P\subseteq \mathbb{R}$ a set satisfying certain tameness conditions, we discuss under which conditions $(\mathcal R,P)$ defines total linear functions that are not definable in \mathcal{R}. Examples of such structures that does define new total linear functions include the cases when $\mathcal{R}$ is a reduct of $(\mathbb{R},<,+,\cdot_{\upharpoonright (0,1)^2},(x\mapsto \lambda x)_{\lambda\in I\subseteq \mathbb{R}})$, and $P= 2^\mathbb{Z}$, or $P$ is an iteration sequence (for any $I$) or $P=\mathbb{Z}$, for $I=\mathbb{Q}$.