*Accepted Paper*

**Inserted:** 25 jun 2019

**Last Updated:** 25 jun 2019

**Journal:** Proceedings of the American Mathematical Society

**Year:** 2018

10 pages

**Abstract:**

We give a simple criterion on the set of probability tangent measures $\mathrm{Tan}(\mu,x)$ of a positive Radon measure $\mu$, which yields lower bounds on the Hausdorff dimension of $\mu$. As an application, we give an elementary and purely algebraic proof of the sharp Hausdorff dimension lower bounds for first-order linear PDE-constrained measures; bounds for closed (measure) differential forms and normal currents are further discussed. A weak structure theorem in the spirit of Ann. Math. 184(3) (2016), pp. 1017-1039 is also discussed for such measures.

**Keywords:**
Hausdorff dimension, Normal current, $\mathcal A$-free measure, structure theorem, PDE constraint, tangent measure