ON SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS AND HAUSDORFF DIMENSIONS

For any [Formula: see text], let the infinite series [Formula: see text] be the Engel expansion of [Formula: see text]. Suppose [Formula: see text] is a strictly increasing function with [Formula: see text] and let [Formula: see text], [Formula: see text] and [Formula: see text] be defined as the sets of numbers [Formula: see text] for which the limit, upper limit and lower limit of [Formula: see text] is equal to [Formula: see text]. In this paper, we qualify the size of the set [Formula: see text], [Formula: see text] and [Formula: see text] in the sense of Hausdorff dimension and show that these three dimensions can be different.

SLOW GROWTH RATE OF THE DIGITS IN ENGEL EXPANSIONS

We are concerned with the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] is the digit of the Engel expansion of [Formula: see text] and [Formula: see text] is a function such that [Formula: see text] as [Formula: see text]. The Hausdorff dimension of [Formula: see text] is studied by Lü and Liu [Hausdorff dimensions of some exceptional sets in Engel expansions, J. Number Theory 185 (2018) 490–498] under the condition that [Formula: see text] grows to infinity. The aim of this paper is to determine the Hausdorff dimension of [Formula: see text] when [Formula: see text] 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 [Formula: see text] and the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] with the convention [Formula: see text].