On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic

TL;DR

This paper introduces the Conservative Matrix Field (CMF), a high-dimensional extension of Apéry limits, connecting classical irrationality proofs with new arithmetic and dynamical phenomena, and proposing conjectures for future research.

Contribution

The paper develops the CMF framework, linking classical Apéry limits to high-dimensional settings and gauge transformations, enabling new approaches to irrationality proofs.

Findings

Numerical experiments reveal unexpected arithmetic phenomena.

Conjectures extend Poincaré--Perron asymptotics to higher dimensions.

Potential for optimization-based irrationality proofs.

Abstract

Ratios of D-finite sequences and their limits -- known as Ap\'ery limits -- have driven much of the work on irrationality proofs since Ap\'ery's 1979 breakthrough proof of the irrationality of $\zeta(3)$. We extend ratios of D-finite sequences to a high-dimensional setting by introducing the Conservative Matrix Field (CMF). We demonstrate how classical Ap\'ery limits are included by this object as special cases. A useful construction of CMFs is provided, drawing a connection to gauge transformations and to representations of shift operators in finite dimensional modules of Ore algebras. Finally, numerical experiments on these objects reveal surprising arithmetic and dynamical phenomena, which are formulated into conjectures. If established, these conjectures would extend Poincar\'e--Perron asymptotics to higher dimensions, potentially opening the door to optimization-based searches for new irrationality proofs.