About one property of entropy of decreasing sequence of measureble partitions

This work is devoted to improving of the assimptotic that appeares in the difinition of the entropy invariant of decreasing sequences of measurable partitions, introduced by A. Vershik. An example of sets that have no good approcsimation by cillinders is represented:
$$ \forall\alpha_n\rightarrow 0\,\exists\gamma_0=\{A,cA\}\;\text{and}\;n_k\rightarrow\infty:H(\gamma^{n_k})>\alpha_{n_k}2^{n_k}. $$