The fractional dimensional theory in Lüroth expansion

It is well known that every x ∈ (0, 1] can be expanded to an infinite Lüroth series in the form of x = 1 d_1(x) + ... + 1 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_φ = { x ∈ (0,1]:d_n(x) ≥φ (n),∀ n ≥ 1} are completely determined, where φ is an integer-valued function defined on ℕ, and φ ( n ) → ∞ as n → ∞.