On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic

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.