The fractional dimensional theory in Lüroth expansion

Abstract

It is well known that every x ∈ (0, 1] can be expanded to an infinite Lüroth series in the form of $$x = {1 \over {{d_1}(x)}} + ... + {1 \over {{d_1}(x)({d_1}(x) - 1...{d_{n - 1}}(x) - 1){d_n}(x)}} + ...,$$ where d n (x) ⩾ 2 for all n ⩾ 1. In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets $${F_\varphi } = \{ x \in (0,1]:{d_n}(x) \ge \varphi (n),\forall n \ge 1\} $$ are completely determined, where φ is an integer-valued function defined on ℕ, and φ(n) → ∞ as n → ∞.