Upper Bounds in the Gap-Ratio Regime for Lacunary Base-M Expansions

Paper3 of a 3 part series starting with "Explicit Transcendental Numbers with Arbitrarily Large Finite Irrationality Measure" This paper contributes: a continued-fraction shadow theorem for bounded-digit lacunary base-M expansions in the finite-ratio regime gₙ₊₁ / gₙ → s > 2; a precise asymptotic formula for the successor partial quotients and successor denominators attached to the truncation convergents; a reduction of the sharp upper bound μ(α) ≤ s to a concrete convergent-growth condition; a first proved equality range, namely μ(α) = s for all s ≥ (3 + √5) / 2 = φ²; an explicit identification of the exact unresolved range for the present method, 2 < s < φ². This paper studies upper bounds for irrationality exponents in lacunary base-M expansions of the form α = ∑n≥1 aₙ / Mgₙ, where M ≥ 2, the digits aₙ are bounded positive integers, and the support sequence (gₙ) is strictly increasing with gap ratio gₙ₊₁ / gₙ → s > 2. Earlier work established the lower bound μ(α) ≥ s by showing that the natural truncation approximants have asymptotic approximation exponent gₙ₊₁ / gₙ. The present paper begins the reverse direction. The paper proves that, when s > 2, the natural truncations are eventually genuine continued-fraction convergents of α. It further shows that the successor partial quotient after the Nth truncation convergent has precise exponential size am_N+1 ≍ Mgₙ₊₁ − 2gₙ, and that the successor denominator satisfies 1 + log Qm_N+1 / log Qm_N → s. Thus the truncation subsequence already exhibits exactly the continued-fraction growth predicted by the conjectural identity μ(α) = s. A central contribution of the paper is a reduction of the sharp upper bound μ(α) ≤ s to a continued-fraction growth condition on all sufficiently large convergents. In this way, the upper-bound problem is converted into a precise exclusion problem for non-truncation convergents. The paper then proves a first nontrivial upper-bound theorem: μ(α) ≤ max(s, 1 + s / (s − 1)), and consequently establishes the exact equality μ(α) = s throughout the range s ≥ (3 + √5) / 2 = φ². This identifies a sharp frontier for the present method and isolates the remaining unresolved interval 2 < s < (3 + √5) / 2. Overall, the paper develops a continued-fraction upper-bound framework for bounded-digit lacunary base-M expansions, proves that the truncation subsequence is asymptotically exact on the continued-fraction side, and obtains the first equality range for the finite-ratio upper-bound problem in this setting.