From ABC to Effective Roth and Ridout Constants for Cubic Roots

Technical Summary: From ABC to Effective Roth and Ridout Constants for Cubic Roots

Problem Statement
Roth's theorem (1955) establishes that for any algebraic number $\alpha$ and any $\epsilon > 0$, there exists a constant $C(\alpha, \epsilon)$ such that $|\alpha - p/q| > C/q^{2+\epsilon}$ for all integers $p, q$. However, the original proof is ineffective, meaning it guarantees the existence of $C$ without providing a method to compute an explicit value. While Bombieri (1994) and Van Frankenhuysen (1999) demonstrated that the ABC conjecture implies Roth's theorem, these implications have not yielded explicit constants in a natural family of cases. This paper addresses the gap between the conditional existence of Roth-type bounds and the derivation of explicit, effective constants, specifically for cubic roots ($\sqrt[3]{k}$), by leveraging an effective form of the ABC conjecture.

Methodology
The authors employ a constructive approach linking Diophantine approximation data with the arithmetic properties of ABC triples. The core methodology involves three main components:

1. Bombieri–Van der Poorten Formula: The authors utilize the explicit formula for the coefficients ($b_n$) of the regular continued fraction of algebraic numbers, specialized for cubic roots ($\alpha = \sqrt[3]{k}$). For convergents $p_n/q_n$, the formula relates the next coefficient $b_{n+1}$ to the remainder term $d_n = |p_n^3 - k q_n^3|$. Specifically, for cubic roots, the leading term dominates the remainder, yielding the rigorous inequality:
   $$b_{n+1} \leq \frac{3p_n^2 q_n}{d_n}$$
   This allows the translation of approximation quality (encoded in $b_n$) into arithmetic data ($d_n$).

2. Construction of ABC Triples: The "resulting equations" from the convergents, $p_n^3 = k q_n^3 + d_n$, are transformed into coprime ABC triples. By defining $g_n = \gcd(k q_n^3, d_n) = \gcd(k, p_n^3)$, the authors construct the triple:
   $$\left( \frac{p_n^3}{g_n}, \frac{k q_n^3}{g_n}, \frac{d_n}{g_n} \right)$$
   This ensures the triple consists of coprime integers, satisfying the prerequisites for applying the ABC conjecture.

3. Derivation of Effective Bounds: Assuming an effective form of the ABC conjecture (where a constant $K_\epsilon$ exists such that $c < K_\epsilon \cdot \text{rad}(abc)^{1+\epsilon}$), the authors substitute the constructed triple into the inequality. By combining this with the Bombieri–Van der Poorten bound for $b_{n+1}$, they derive an explicit relationship between the $\epsilon$ parameter in Roth's theorem ($\epsilon_{Roth}$) and the $\epsilon$ in the ABC conjecture ($\epsilon_{ABC}$), as well as an explicit upper bound for the inverse of the Roth constant ($1/C$).

Key Contributions and Results

• Explicit Roth Constants for Cubic Roots: The paper derives a fully explicit bound for the inverse of the Roth constant for $\sqrt[3]{k}$. The relationship between the parameters is given by $\epsilon_{Roth} = \frac{3\epsilon_{ABC}}{1+\epsilon_{ABC}}$. The bound for $1/C$ is expressed in terms of the ABC constant $K_\epsilon$, the integer $k$, and the initial convergents.
• Numerical Analysis for $\sqrt[3]{2}$: The authors apply their method to $\sqrt[3]{2}$. For $\epsilon_{Roth} = 0.5$, the ABC-based derivation yields a bound of approximately $13.16$ for $1/C$, compared to Korobov's exact result of $3.15$. While the ABC-derived bound is less precise in this specific instance (attributed to the use of coarse upper bounds for convergents), it demonstrates the feasibility of the method. Similar calculations are provided for $\epsilon_{Roth} = 0.4$.
• Extension to Ridout's Theorem: The methodology is extended to provide explicit bounds for the number of solutions in Ridout's theorem (concerning approximations by $S$-integers) for both square roots and cubic roots. The authors show that for fixed $\epsilon > 0$ and an effective ABC constant, the set of solutions is finite and explicitly bounded.
• Approximation Gain and the Conjecture: A new concept, "approximation gain," is introduced as a refinement of the "quality" of an ABC hit. It is defined based on the logarithmic ratio of the terms in the resulting equation. The authors prove that for cubic roots ($k \geq 3$ and $k=2$), the approximation gain is strictly bounded by $3/2$.
  • Theorem 2.7 & 2.8: For all resulting equations arising from convergents to $\sqrt[3]{k}$, the approximation gain is less than $3/2$.
  • Computational Verification: This bound was verified computationally for $c < 2^{63}$.

Significance and Claims
The paper claims to make the dependence of Roth-type constants on ABC parameters explicit, moving from a theoretical implication to a calculable framework for cubic roots. The significance lies in:

1. Bridging Theory and Computation: It provides a concrete pathway to compute Diophantine approximation bounds assuming an effective ABC conjecture.
2. Strategic Decomposition of ABC: The authors propose a potential strategy for attacking the ABC conjecture by separating the "quality" of an ABC hit into two components: "approximation gain" and "power gain."
   • They conjecture that the approximation gain is always bounded by $3/2$ (proven for cubic roots).
   • They suggest that if the power gain can also be bounded (e.g., by 3), the ABC conjecture could be proven.
3. Refinement of Bounds: The work introduces the approximation gain as a metric that is always lower than or equal to the standard quality, offering a potentially tighter constraint for analyzing ABC hits.

The authors remain modest regarding the precision of the derived Roth constants, noting that in tested cases (like $\sqrt[3]{2}$), the ABC-based bounds are currently looser than known specific results (e.g., Korobov's). However, they emphasize that the approach successfully makes the constants explicit and offers a novel decomposition for future research into the ABC conjecture.