The irrationality exponent of small-height binary lacunary series is 2

Let N ≥ 2 be an integer and let λ_j be nonzero rational numbers with log H(λ_j) = o(2^j), where H(u/v) = max(|u|, v) for u/v in lowest terms with v > 0. We prove that Σj≥0 λ_j N^(−2^j) is irrational and has irrationality exponent 2. The proof uses Padé approximants to the tails Σr≥0 λₖ₊ᵣ z^(2^r). The two Hankel families they need satisfy parity recursions, so they are nonzero for all nonzero weights. After the finite prefix of the series is cleared, the approximants of order n give integer linear forms with height N^((n+1/2)2^k) and error N^(−(n+1/2)2^k), up to factors e^(o(2^k)). Only finitely many orders are used for each target exponent. The height condition cannot be weakened to log H(λ_j) ≤ C·2^j for any constant C > 0: for every C some such weights give exponent greater than 2, and for larger C positive integer weights can give a rational number or a Liouville number.