  We show that the Hausdorff dimension of any proper Teichm\"uller horocycle
flow orbit closure on any $\mathrm{SL}{(2,\mathbf{R})}$-invariant subvariety of
Abelian or quadratic differentials is bounded away from the dimension of the
subvariety in terms of the polynomial mixing rate of the Teichm\"uller
horocycle flow on the subvariety. The proof is based on abstract methods for
measurable flows adapted from work of Bourgain and Katz on sparse ergodic
theorems.
