Gaps in Intervals of N-expansions

For N ∈ N_≥2 and α ∈ R such that 0 < α ≤ √N −1, the continued fraction map T_α : [α,α+1] → [α,α+1) is defined as T_α(x) := N/x−d(x), where d : [α,α+1] → N is defined by d(x) := N/x−α. A maximal open interval (a,b) ⊂ I_α is called a gap of I_α if for almost every x ∈ I_α there is an n₀(_x) ∈ N such that x_n / ∉ (a,b) for all n ≥ n₀. In this paper, all conditions are given in which I_α is gapless. For α = √N−1 it is shown that the number of gaps is a finite, monotonically nondecreasing and unbounded function of N.