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.
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A. M. Vershik, “The asymptotics of the partition of the cube into Weyl simplices, and an encoding of a Bernoulli scheme,” Funkts. Anal. Prilozhen., 53, No. 2, 11–31 (2019).
V. A. Rokhlin, “Lectures on the entropy theory of measure-preserving transformations,” Russian Math. Surveys, 22, No. 5, 1–52 (1967).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 481, 2019, pp. 5–11.
Translated by G. Veprev.
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Veprev, G.A. Estimating the Asymptotic Behavior of the Entropy of an Invariant Sequence of Partitions of the Infinite-Dimensional Cube. J Math Sci 247, 641–645 (2020). https://doi.org/10.1007/s10958-020-04826-w
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DOI: https://doi.org/10.1007/s10958-020-04826-w