The irrationality exponent of small-height lacunary series

Let N ≥ 2 be an integer, let u_1 < u_2 < ⋯ be positive integers with liminf u_{j+1}/u_j ≥ 2, and let λ_j be nonzero rational numbers with log H(λ_j) = o(u_j). Then x = Σ_{j≥1} λ_j N^(−u_j) is irrational and its irrationality exponent is limsup u_{j+1}/u_j, finite or infinite. The proof compares logarithmic gaps between the continued-fraction denominators of consecutive truncations. A perturbation lemma places each gap of the next truncation in at least one of five classes (bounded, inherited, central, short, reflected), each with an explicit error. Reflection moves a large gap far enough to the right to give a uniform bound at the boundary liminf u_{j+1}/u_j = 2. The coefficient condition allows arbitrary signs and cannot be replaced by log H(λ_j) ≤ C·u_j for any fixed C > 0.