ESTIMATING THE ASYMPTOTIC BEHAVIOR OF THE ENTROPY OF AN INVARIANT SEQUENCE OF PARTITIONS OF THE INFINITE-DIMENSIONAL CUBE

摘要

In this paper, we solve the question, posed by A. M. Vershik, about the asymptotic behavior of the entropies of a given sequence of partitions of the infinite-dimensional cube satisfying the invariance and exhaustibility properties. On the one hand, it is proved that the entropy sequence increases faster than a linear function. On the other hand, we construct a series of examples that show that the estimate is sharp: for any given sequence increasing faster than a linear function, the entropy of a sequence of partitions can increase slower than the given sequence.