Geometric Phase Integrals and Irrationality Tests

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Let F(x) be an analytical, real valued function defined on a compact domain B ⊂ R. We prove that the problem of establishing the irrationality of F(x) evaluated at x0 ∈ B can be stated with respect to the convergence of the phase of a suitable integral I(h), defined on an open, bounded domain, for h that goes to infinity. This is derived as a consequence of a similar equivalence, that establishes the existence of isolated solutions of systems equations of analytical functions on compact real domains in Rp, if and only if the phase of a suitable “geometric” complex phase integral I(h) converges for h → ∞. We finally highlight how the method can be easily adapted to be relevant for the study of the existence of rational or integer points on curves in bounded domains, and we sketch some potential theoretical developments of the method.


This is the pre-print of an article that will be published at a later date. This version has not yet undergone peer review and may differ from the final, published version.