Technology Category
Knowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingMachine Learning: Matrix & Tensor MethodsConstraint Satisfaction and Optimization: Satisfiability Modulo Theories
On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic
🤖 AI Summary
This project generalizes the Apéry limit (i.e., the ratio limit) of D-finite sequences and its underlying Poincaré–Perron asymptotic theory to higher dimensions, aiming to open new avenues for irrationality proofs. Method: We introduce the novel concept of *conservative matrix fields* (CMFs) to uniformly characterize the asymptotic behavior of multidimensional ratio sequences, establishing intrinsic connections with shift-operator representations in Ore algebras and gauge transformations. Our approach integrates D-finite sequence analysis, modular representation theory, analytic asymptotics, and high-precision numerical experimentation. Contributions: (i) the first natural higher-dimensional generalization of Apéry limits; (ii) unification of classical asymptotic theory within a coherent framework; (iii) discovery of new arithmetic dynamical phenomena; and (iv) formulation of fundamental conjectures on multidimensional asymptotic stability and irrationality—providing both theoretical tools and computational paradigms for constructing novel irrationality proofs.
Technology Category
Application Category
📝 Abstract
Ratios of D-finite sequences and their limits -- known as Apéry limits -- have driven much of the work on irrationality proofs since Apéry's 1979 breakthrough proof of the irrationality of $ζ(3)$. We extend ratios of D-finite sequences to a high-dimensional setting by introducing the Conservative Matrix Field (CMF). We demonstrate how classical Apéry 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 Poincaré--Perron asymptotics to higher dimensions, potentially opening the door to optimization-based searches for new irrationality proofs.
Problem
Research questions and friction points this paper is trying to address.
Extend D-finite sequence ratios to high-dimensional Conservative Matrix Fields
Connect CMFs to gauge transformations and Ore algebra representations
Explore arithmetic and dynamical phenomena for new irrationality proofs
Innovation
Methods, ideas, or system contributions that make the work stand out.
Introducing Conservative Matrix Field (CMF)
Connecting CMFs to gauge transformations
Extending Poincaré--Perron asymptotics dimensions
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