  Let K/Q be a field extension of finite degree and let P(t) be a polynomial
over Q that splits into linear factors over Q. We show that any smooth model of
the affine variety defined by the equation N_{K/Q} (k) = P(t) satisfies the
Hasse principle and weak approximation whenever the Brauer-Manin obstruction is
empty. Our proof is based on a combination of methods from additive
combinatorics due to Green-Tao and Green-Tao-Ziegler, together with an
application of the descent theory of Colliot-Th\'el\`ene and Sansuc.
