  Let $F(x)$ be an analytical, real valued function defined on a compact domain
$\mathcal {B}\subset\mathbb{R}$. We prove that the problem of establishing the
irrationality of $F(x)$ evaluated at $x_0\in \mathcal{B}$ 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 $\mathbb{R}^p$, if and only if the phase of a suitable ``geometric''
complex phase integral $I(h)$ converges for $h\rightarrow \infty$. 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.
