From ABC to Effective Roth and Ridout Constants for Cubic Roots
Karsten Müller, Michael Taktikos
TL;DR
The paper develops explicit, effective bounds for Roth-type Diophantine approximation in the cubic-root setting by assuming an effective form of the ABC conjecture. Leveraging the Bombieri–Van der Poorten explicit continued-fraction formula, it connects ABC data with convergents of cubic irrationalities to derive an explicit bound on the inverse Roth constant and an explicit Ridout-type bound. It introduces the notion of approximation gain and shows that, for cubic roots, the approximation gain can be bounded by $\tfrac{3}{2}$, while separating power gains highlights the need for stronger ABC results. The work provides concrete demonstrations on $\sqrt[3]{2}$ and outlines a framework for refining ABC to attack broader Diophantine problems. The approach positions a path toward effective, computable constants in Diophantine approximation through explicit ABC bounds and continued-fraction analysis.
Abstract
Enrico Bombieri showed conditionally (1994) that the ABC conjecture implies Roth's theorem, and Van Frankenhuysen (1999) later provided a complete proof. Building on Bombieri's and Van der Poorten's explicit formula for continued-fraction coefficients of algebraic numbers (specialized to cubic roots) we derive an effective bound for a Roth-type constant assuming an effective form of ABC. Roth's original argument establishes existence but does not yield an explicit value; our approach makes the dependence on the ABC parameters explicit and also gives an explicit bound in the corresponding special case of Ridout's theorem. We then introduce the notion of approximation gain as a refinement of the quality of an abc-triple. For c in a large computational range, the approximation gain remains below a strikingly small threshold, motivating the conjecture that the approximation gain is always smaller than 1.5. This suggests a potential strategy for attacking ABC by bounding approximation gain and power gain separately.