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 p_N/q_N satisfy the asymptotic law −log|α − p_N/q_N| / log q_N = 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 < s < ∞, the construction yields the lower bound μ(α) ≥ s; when gₙ₊₁/gₙ → ∞, it yields Liouville numbers. In particular, exponential gaps gₙ = ⌊sⁿ⌋ produce explicit transcendental numbers for every s > 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.