In this work we extend many classical results concerning the relationship
between densities, tangents and rectifiability to the parabolic spaces, namely
Rn+1 equipped with parabolic dilations. In particular we prove a
Marstrand-Mattila rectifiability criterion for measures of general dimension,
we provide a characterisation through densities of intrinsic rectifiable
measures, and we study the structure of 1-codimensional uniform measures.
Finally, we apply some of our results to the study of a quantitative version of
parabolic rectifiability: we prove that the weak constant density condition for
a 1-codimensional Ahlfors-regular measure implies the bilateral weak
geometric lemma