Geodesic
Estimating the asymptotic behavior of the entropy of an invariant sequence of partitions of the infinite-dimensional cube
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 5-11

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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. 
@article{ZNSL_2019_481_a0,
     author = {G. A. Veprev},
     title = {Estimating the asymptotic behavior of the entropy of an invariant sequence of partitions of the infinite-dimensional cube},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--11},
     publisher = {mathdoc},
     volume = {481},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_481_a0/}
}
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G. A. Veprev. Estimating the asymptotic behavior of the entropy of an invariant sequence of partitions of the infinite-dimensional cube. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 5-11. http://geodesic.mathdoc.fr/item/ZNSL_2019_481_a0/