  We refine two results in the paper entitled "Sofic mean dimension" by Hanfeng
Li, improving two inequalities with two equalities, respectively, for sofic
mean dimension of typical actions. On the one hand, we study sofic mean
dimension of full shifts, for which, Li provided an upper bound which however
is not optimal. We prove a more delicate estimate from above, which is optimal
for sofic mean dimension of full shifts over arbitrary alphabets (i.e. compact
metrizable spaces). Our refinement, together with the techniques (in relation
to an estimate from below) in the paper entitled "Mean dimension of full
shifts" by Masaki Tsukamoto, eventually allows us to get the exact value of
sofic mean dimension of full shifts over any finite dimensional compact
metrizable spaces. On the other hand, we investigate finite group actions. In
contrast to the case that the acting group is infinite (and amenable), Li
showed that if a finite group acts continuously on a finite dimensional compact
metrizable space, then sofic mean dimension may be different from (strictly
less than) the classical (i.e. amenable) mean dimension (an explicitly known
value in this case). We strengthen this result by proving a sharp lower bound,
which, combining with the upper bound, gives the exact value of sofic mean
dimension for all the actions of finite groups on finite dimensional compact
metrizable spaces. Furthermore, this equality leads to a satisfactory
comparison theorem for those actions, deciding when sofic mean dimension would
coincide with classical mean dimension. Moreover, our two results, in
particular, verify for a typical class of sofic group actions that sofic mean
dimension does not depend on sofic approximation sequences.
