A variational principle for metric mean dimension via lower Brin-Katok local entropy

We prove a finite-scale comparison between lower Brin-Katok local entropy and Katok covering entropy. Let $(\mathcal{X},d,T)$ be a compact metric topological dynamical system and let $\mu$ be ergodic. Then, for every $\epsilon>0$ and every $\delta\in(0,1)$, $$ h^K_\mu(6\epsilon,\delta)\leq \underline h^{BK}_\mu(\epsilon). $$ Combining this estimate with the usual Katok-type variational principle for metric mean dimension gives the corresponding variational principle with lower Brin-Katok local entropy.