Explicit Transcendental Numbers with Arbitrarily Large Finite Irrationality Measure

This paper develops a gap-ratio principle for lacunary base-M expansions of the form α = ∑ₙ≥1 aₙ / Mᵍⁿ, where the digits aₙ are bounded positive integers and the exponents gₙ form a strictly increasing sequence. The main theorem shows that the natural truncation approximants pN/qN satisfy the asymptotic law −log|α − pN/qN| / log qN = gₙ₊₁/gₙ + O (1/gₙ), so the approximation behavior is governed by the growth law of the gap sequence. As consequences, when gₙ₊₁/gₙ → s with 1 2, while factorial and power-factorial gaps produce a Liouville regime. The paper also proves that the exponential-gap family has Hausdorff dimension 0, and includes numerical experiments based on true truncation errors showing strong agreement with the gap-ratio law across exponential, factorial, and power-factorial examples.