SLOW GROWTH RATE OF THE DIGITS IN ENGEL EXPANSIONS

We are concerned with 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 and phi : N -> R+ is a function such that phi(n) -> infinity as n -> infinity. The Hausdorff dimension of E-phi is studied by Lu and Liu [Hausdorff dimensions of some exceptional sets in Engel expansions, J. Number Theory 185 (2018) 490-498] under the condition that phi(n + 1) - phi(n) grows to infinity. The aim of this paper is to determine the Hausdorff dimension of E-phi, when phi(n) slowly increases to infinity, such as in logarithmic functions and power functions with small exponents. We also provide a detailed analysis of the gaps between consecutive digits. This includes the central limit theorem and law of the iterated logarithm for log Delta(n) and the Hausdorff dimension of the set F phi = {x is an element of (0,1] : lim(n ->infinity) log Delta(n)(x)/phi(n) = 1}, where Delta(n) (x) := d(n)(x) - d(n-1)(x) with the convention d(0)(x) 0.