On the exact rate of convergence of digits in Engel expansions

Let phi : N -> R+ be a function satisfying phi(n) -> infinity as n -> infinity. Denote by < d(1)(x), d(2)(x), ..., d(n)(x), ...> the Engel expansion of an irrational number x is an element of(0, 1). We study the Baire classification and Hausdorff dimension of D-phi (alpha, beta) := (x is an element of(0, 1)\ Q : lim inf(n ->infinity) log d(n)(x) - n/phi(n) = alpha, lim sup(n ->infinity) log d(n) (x) - n/phi(n) = beta} for alpha, beta is an element of[-infinity, infinity] with alpha <= beta. Under the condition that phi(n + 1) - phi(n) -> 0 as n -> infinity, we show that the set D-phi(alpha, beta) is residual if and only if alpha = -infinity and beta = infinity. Under the conditions that phi is increasing and phi(n + 1) - phi(n) -> 0 as n -> infinity, we prove that D-phi(alpha, beta) has full Hausdorff dimension for any -infinity <= alpha <= beta <= infinity, which improves the result of Liu and Wu (2003). We also do some similar analyses for the gap of consecutive digits. (c) 2023 Elsevier Inc. All rights reserved.