Localized growth speed of the digits in Engel expansions

We introduce a localized version of a Galambos's question about the growth speed of the digits in Engel expansion, namely the set{ 1 } x & ISIN; (0, 1] : lim nlog dn(x) = & alpha;(x) , n & RARR;& INFIN;where & alpha; : [0, 1] & RARR; [0, & INFIN;] is a nonnegative continuous function and dn(x) denotes the nth digits in the Engel expansion of x. The Hausdorff dimension is shown to be irrelevant of the function & alpha;(x). As applications, this answers Galambos's question and strengthens some results by Liu & Wu.& COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

We introduce a localized version of Galambos's question about the growth speed of the digits in Engel expansion and show that the Hausdorff dimension is irrelevant of the function & alpha;(x).