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In this note we extend the dimension theory for the spaces$\widetilde{\mathcal H}_g^k(\mu)$ of twisted $k$-differentials defined byFarkas and Pandharipande in [FP15] to the case $k>1$. In particular, we showthat the intersection $\widetilde{\mathcal H}_g^k(\mu) \cap \mathcal{M}_{g,n}$is a union of smooth components of the expected dimensions for all $k\geq 0$.We also extend a conjectural formula from [FP15] for a weighted fundamentalclass of $\widetilde{\mathcal H}_g^k(\mu)$ and provide evidence in low genus.

Dimension theory of the moduli space of twisted $k$-differentials