Arithmetic Holonomy Bounds

• Arithmetic holonomy bounds are quantitative inequalities that connect analytic continuation of holonomic functions with precise arithmetic properties in transcendental number theory.
• They enable algorithmic resolution of classical Diophantine equations by controlling denominator growth and leveraging overconvergence techniques.
• The framework integrates analytic continuation, arithmetic combinatorics, and modular monodromy to derive explicit irrationality measures and effective height bounds.

Arithmetic holonomy bounds are quantitative inequalities governing the analytic continuation (monodromy) and algebraic independence properties of solutions to certain holonomic (differentially algebraic) functions, specifically as they arise in Apéry-type constructions and related settings in transcendental number theory. These bounds are leveraged to obtain effective results in Diophantine approximation and transcendence, including explicit irrationality measures, effective height bounds, and algorithmic solutions to classical equations such as the S-unit, Thue–Mahler, and superelliptic equations. The framework integrates methods from analytic continuation, arithmetic combinatorics, and modular/hypergeometric monodromy, and may be viewed as a multivalent analytic generalization of classical approaches due to Thue, Siegel, Baker, and Bombieri.

1. Quantitative Arithmetic Holonomy Bounds

The primary technical innovation is the establishment of inequalities for collections of holonomic power series: $$f_i(x) = \sum_{n=0}^{\infty} a_{i,n} \frac{x^n}{\prod_{j=1}^{r_i} [1,\ldots,b_{i,j} n]}, \qquad a_{i,n} \in \mathbb{Z}$$ where the sequence $(b_{i,j})$ encodes the denominator growth ("denominator type", Editor's term). The quantitative holonomy bound is formulated with respect to a holomorphic mapping $\varphi: (\Delta, 0) \to (\mathbb{C}, 0)$ (with conformal size $\log|\varphi'(0)|$) and an average $\tau(\mathbf{b})$ determined by the denominator types.

If $m$ linearly independent functions admit analytic continuation ("overconverge") under pullback by $\varphi^*$, the holonomy bound reads: $$m \leq \frac{\iint_{t^2} \log |\varphi(z) - \varphi(w)|\,dz\,dw}{\log|\varphi'(0)| - \tau(\mathbf{b})}$$ A refined version for Apéry limits improves this by including contributions from irrationality measures $\mu(\eta)$ and a convergence parameter $\rho$: $$m \leq \frac{\iint_{t^2} \log |\varphi(z) - \varphi(w)|\,dz\,dw}{\log|\varphi'(0)| - \tau(\mathbf{b}) - (1-\gamma)\left(\frac{2}{\mu(\eta)} - \frac{1-\gamma}{\mu(\eta)^2}\right)\log(1/\rho)}$$ Here, $\gamma$ captures the proportion of functions related to a given irrationality parameter.

These bounds are applied to spans of functions of the form $g(x) = B(x) - \eta A(x)$, where $\eta$ is an Apéry-type limit and denominator types are explicitly managed. The analytic control (encoded in the integral numerator and conformal size) directly restricts the possible dimensions and effective arithmetic properties of the function space.

2. Effective Diophantine Approximation via Holonomy Bounds

A main application is in bounding solutions to multiplicative Diophantine inequalities. Given a finitely generated subgroup $\Gamma \subset K^\times$ of a number field $K$, a place $v$, and $\varepsilon > 0$, the paper proves: $$|1 - A\gamma|_v \leq H(\gamma)^{-\varepsilon} \implies h(\gamma) \leq C(K, v, \Gamma, \varepsilon)(1 + h(A))$$ for all $A \in K^\times$, where $h(\gamma)$ is a height function and $C$ is explicit.

This bound is derived by constructing holonomic functions tailored to the arithmetic and analytic properties of $A$ and $\Gamma$, and invoking the quantitative holonomy inequalities described above. Classical approaches (linear forms in logarithms, Bombieri's geometry-of-numbers reduction) are thus reinterpreted through analytic continuation and monodromy of specifically constructed power series.

Using this machinery, a host of equations—including the S-unit equation $x + y = 1$ ($x, y \in O_{K,S}^\times$), Thue–Mahler equations $F(x, y) = \pm 1$, and hyperelliptic/superelliptic equations $y^m = f(x)$—can be algorithmically solved in the sense that the search space for solutions is effectively bounded.

3. Comparison with Classical Methods

The method generalizes the classical Thue–Siegel–Baker hypergeometric approach and offers an alternate mechanism to Bombieri’s equivariant Thue–Siegel method. Rather than focusing on the existence of auxiliary functions or intricate transcendence measures from linear forms in logarithms, the new framework exploits explicit analytic continuation ("multivalent continuation") and manages denominator structures via an "algebraic dihedral construction".

Key steps include:

• Building Apéry-type generating functions with well-controlled denominator growth,
• Pullback via analytic maps with large conformal size to induce overconvergence,
• Quantifying monodromy/holonomy properties,
• Translating analytic inequalities directly into arithmetic bounds.

A prototypical analytic identity used is

$$\sum_{m, n=0}^{\infty} {}_2F_1(-\nu-n, -m; -m-n; x) \binom{m+n}{m} y^m z^n = \frac{(1 - xy)^\nu}{1 - y - z + xyz}$$

which, under suitable specializations, yields generating functions with finite dihedral monodromy—facilitating explicit control of their analytic and arithmetic behavior.

4. Concrete Applications and Advanced Results

Specific consequences presented include:

• Algorithmic solution of Diophantine equations: Height bounds derived from holonomy inequalities yield finite search spaces for S-unit, Thue–Mahler, and related equations.
• Transcendence of $\pi$: Apéry-type approximants for the logarithm are used. Functions $H(z) = B(z) - \pi A(z)$, with explicit denominator type, yield via the holonomy bounds a contradiction to the algebraicity of $\pi$, providing new transcendence proofs.
• Irrationality measures for $L(2, \chi_{-3})$ and $2$-adic $\zeta(5)$: By refining the holonomy bounds to encapsulate effective irrationality exponents $\mu(\eta)$, the paper provides explicit lower bounds of the form $|L(2, \chi_{-3}) - p/q| > 1/q^{24781}$ (modulo a finite set of exceptions) and $|\zeta_2(5) - p/q|_2 > 1/\max(|p|,|q|)^{20}$.
• p-adic analogues: The method applies uniformly to p-adic periods via overconvergent modular forms, using q-expansions associated to modular curves and solutions of linear differential equations.
• Power sharpening of Liouville’s theorem: Effective irrationality measures for high-order roots of algebraic numbers are derived without requiring deep input from linear forms in logarithms.

5. Theoretical and Computational Impact

Arithmetic holonomy bounds provide a unified analytic–arithmetic framework that:

• Supplies direct, quantitative links between analytic continuation (multivalency, monodromy) and Diophantine bounds,
• Enables the computation of explicit irrationality exponents,
• Offers algorithmic, explicit control in settings previously studied only by qualitative or non-effective means,
• Treats real, complex, and p-adic contexts simultaneously.

The approach supersedes certain classical transcendence and irrationality methods for key classes of numbers (modular periods, special $L$-values, $p$-adic zeta constants, etc.), and yields new practical tools for bounding Diophantine subsets.

6. Broader Context and Future Directions

The framework is extensible; future developments promised in the paper will integrate broader classes of holonomic functions and modular settings. The interplay between analytic monodromy, denominator structure, and arithmetic independence suggests unresolved questions regarding optimality, universality, and the analytic geometry of moduli spaces of Diophantine constraints. This method offers promising avenues in explicit algebraic independence results, effective enumeration of Diophantine solutions, and the transcendence theory of periods and $L$-values.