科研之友
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摘要
Let phi : N -> R+ be a non-decreasing function such that phi(n)/log n -> infinity as n -> infinity. In this paper, we completely determine the Hausdorff dimension of the set @@@ E-phi = {x is an element of (0, 1]: lim(n ->infinity) log d(n)(x)/phi(n) = 1}, @@@ where d(n)(x) is the digit of the Engel expansion of x. It significantly generalises the existing results on the Hausdorff dimension of E-phi. Besides, some analogous results for gaps and ratios of consecutive digits are also provided.
