Arithmetic holonomy bounds and effective Diophantine approximation
Frank Calegari, Vesselin Dimitrov, Yunqing Tang
TL;DR
The paper develops a novel framework of arithmetic holonomy bounds to obtain effective Diophantine approximations on $\mathbf{G}_m$ and related groups, integrating Apéry-limit techniques with a quantitative, slope-based approach. A dihedral hypergeometric construction is then employed to derive explicit irrationality measures for high order roots and to prove transcendence-type results for $\pi$, alongside irrationality results for special $L$-values and a $p$-adic zeta value. The results unify Apéry-type bounds, Thue–Siegel–Baker methods, and dihedral algebraic structures to produce effective, algorithmic Diophantine content, with several concrete numerical irrationality exponents demonstrated. The authors indicate that full development of explicit constants and extensions to multivariable $S$-unit equations will appear in future work, signaling a broad program of effective Diophantine analysis grounded in holonomy and multivalued analytic techniques.
Abstract
In this paper, we explore several threads arising from our recent joint work on arithmetic holonomy bounds, which were originally devised to prove new irrationality results based on the method of Apéry limits. We propose a new method to address effective Diophantine approximation on the projective line and the multiplicative group. This method, and all our other results in the paper, emerged from quantifying our holonomy bounds in a way that directly yields effective measures of irrationality and linear independence. Applying these to a dihedral algebraic construction, we derive good effective irrationality measures for high order roots of an algebraic number, in an approach that might be considered a multivalent continuation of the classical hypergeometric method of Thue, Siegel, and Baker. A well-known Dirichlet approximation argument of Bombieri allows one to derive from this the classical effective Diophantine theorems, hitherto only approachable by Baker's linear forms in logarithms or by Bombieri's equivariant Thue--Siegel method. These include the algorithmic resolution of the two-variable $S$-unit equation, the Thue--Mahler equation, and the hyperelliptic and superelliptic equations, as well as the Baker--Feldman effective power sharpening of Liouville's theorem. We also give some other applications, including irrationality measures for the classical $L(2,χ_{-3})$ and the $2$-adic $ζ(5)$, and a new proof of the transcendence of $π$. Due to space limitations, a full development of these ideas will be deferred to future work.