Abstract
The present paper is concerned with the Hausdorff dimension of certain sets arising in Engel expansions. In particular, the Hausdorff dimension of the set is completely determined, where An(x) can stand for the digit, gap and ratio between two consecutive digits in the Engel expansion of x and ϕ is a positive function defined on natural numbers. These results significantly extend the existing results of Galambos’ open problems on the Hausdorff dimension of sets related to the growth rate of digits.
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