$\vartheta$-dependent terms does not make much sense in the scope of the Newtonian framework, because for faster scattering processes there would be significant general relativistic corrections. As Newtonian gravity is the lowest order correction of SR coming from GR, we have obtained a condition for DSR theories to satisfy the correspondence principle in its full meaning, namely, the lowest-order effects of DSR must be quadratic. DSR 2 for example, proposed in [@MS], with linear corrections would be at odds with it.

One important difference of the present approach to “full" DSR is the use of the free energy-momentum relations for both kinds of variables, rather than of modified ones for $E$ and $p$. This important aspect of DSR does not show up in the present calculations, because in the considered approximations the mass term does not play a role, and $E\approx p$ as well as $\varepsilon\approx\pi$, the calculations were essentially carried out for momenta alone.

The above considerations can easily be applied to higher dimensions. In $d>3$ space dimensions the Planck mass $m_P^{(d)}$ is by orders of magnitude smaller, on the other hand, the Newtonian potential goes as $r^{2-d}$. See for example [@G]. In consequence, the lowest order correction of 