the scattering angle behaves as $$\frac{p^{d-1}}{\left(m_P^{(d)}\right)^{d-1}},$$ if there are compactified dimensions, large enough for classical gravity to be a reasonable approximation when the minimal distance is as small as the magnitude of these dimensions. In consequence, in these cases the lowest-order corrections of DSR must be of order $d-1$, when compatibility with classical gravity is desired.

Comparison with GUP
===================

This approach is characterized by making a principal distinction between the energy and momentum $(E,\vec p\,)$ of a particle and its associated frequency and wave vector $(\omega,\vec k\,)$. Their relation is most generally written as $$\label{Efp}
(\omega,\vec k\,)=\left(E\cdot f(E,\vec p\,),\vec p\cdot g(E,\vec
p\,)\right).$$ Energy and momentum are assumed to be unbounded, whereas $\omega$ and $\vec k$ are bounded by orders of magnitude 1/[*Planck time*]{} and 1/[*Planck length*]{}, respectively. The functions $f$ and $g$ can be chosen analogously to the functions relating physical and pseudo-variables in arbitrary versions of DSR. Nevertheless, the interpretation is different: There are no merely auxiliary variables, both $(E,\vec p\,)$ and $(\omega,\vec k\,)$ have a clear physical meaning and there is a natural relation between momentum and position space from the beginning.

Comparing (\[Efp\]) with the standard relation $$(E,\vec p\,)=\hbar(\omega,\vec k\,),$$ one finds energy-momentum dependent constants $$\tilde{\hbar}(E,\vec 