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We compute a lower bound of Billingsley–Hausdorff dimension, defined by Gibbs measure, of the level set related to Birkhoff average in the countable symbolic spaceNN. 2020 Mathematics Subject Classification. 28A80, 37A05, 37A35, 37B10, 37C45. Manuscript received 2nd July 2019, revised 14th February 2020, accepted 18th February 2020.

On the Billingsley dimension of Birkhoff average in the countable symbolic space