MULTIFRACTAL ANALYSIS OF THE DIVERGENCE POINTS ASSOCIATED WITH THE GROWTH OF DIGITS IN ENGEL EXPANSIONS

In this paper, we are concerned with the multifractal analysis of the divergence points in Engel expansions. Let [Formula: see text] be an irrational number with Engel expansion [Formula: see text]. For any [Formula: see text], let [Formula: see text] We prove that the Hausdorff dimension of [Formula: see text] is [Formula: see text] when [Formula: see text], and it is zero when [Formula: see text]. This indicates that the Hausdorff dimension of [Formula: see text] is independent of [Formula: see text]. A very different phenomenon is shown for the gap of consecutive digits. For any irrational number [Formula: see text] and [Formula: see text], let [Formula: see text] with [Formula: see text]. We derive that, for any [Formula: see text], the set [Formula: see text] has Hausdorff dimension [Formula: see text].