Weak-field limit of $f(R)$ gravity in three and more spatial dimensions

We investigate a pointlike massive source in nonlinear $f(R)$ theories in the case of an arbitrary number of spatial dimensions $D\ge 3$. If $D>3$ then extra dimensions undergo toroidal compactification. We consider a weak-field approximation with Minkowski and de Sitter background solutions. In both these cases pointlike massive sources demonstrate good agreement with experimental data only in the case of ordinary three-dimensional ($D=3$) space. We generalize this result to the case of a perfect fluid with dustlike equations of state in the external and internal spaces. This perfect fluid is uniformly smeared over all extra dimensions and enclosed in a three-dimensional sphere. In ordinary three-dimensional ($D=3$) space, our formulas are useful for experimental constraints on parameters of $f(R)$ models.