On the Density Problem in the Parabolic Space

In this work we establish relationships between densities, tangents, and rectifiability in parabolic spaces, namely R n + 1 \mathbb {R}^{n+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 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 1 -codimensional Ahlfors-regular measure implies the bilateral weak geometric lemma.