New Directions in Irrationality

Abstract

Since Apéry's proof of the irrationality of $ζ(3)$, decades of work have explored integrals, hypergeometric identities, and recurrences with exceptional arithmetic properties, to discover the strongest quantitative bounds on irrationality measures for zeta values and related constants. Yet, most findings have relied on specially tailored families of such constructions, making their systematic extension difficult. Here we show how these constructions can be connected and generalized into a common multidimensional structure based on conservative matrix fields (CMFs), enabling a systematic discovery of new irrationality proofs and quantitative bounds for irrationality exponents. CMFs are finite-dimensional cocycles of compatible rational matrix shifts, and are shown to capture all the classical constructions developed since Apéry as one-dimensional trajectories. The initial point encodes the mathematical constants; varying the trajectory within a suitable convergence region produces different approximation schemes for the same constants. We use this structure to improve the irrationality measures of famous constants, yielding for example $μ(ζ(2))<4.62675$ and $μ(ζ(3))<5.4197$, and proving that $1,ζ(3),ζ(4)$ are linearly independent over $\mathbb{Q}$, yielding $μ(Aζ(3)+Bζ(4))<66.3$ for every $(A,B)\in\mathbb{Q}^2\setminus\{(0,0)\}$. The same approach demonstrated over other CMFs provides irrationality proving trajectories for infinitely more mathematical constants. We further construct higher-rank CMFs with explicit initial states for $1,ζ(2),ζ(3),\ldots,ζ(d+1)$, and identify the arithmetic conditions needed to obtain further, higher-order independence results.