  To compare continued fraction digits with the denominators of the
corresponding approximants we introduce the arithmetic-geometric scaling. We
will completely determine its multifractal spectrum by means of a number
theoretical free energy function and show that the Hausdorff dimension of sets
consisting of irrationals with the same scaling exponent coincides with the
Legendre transform of this free energy function. Furthermore, we identify the
asymptotic of the local behaviour of the spectrum at the right boundary point
and discuss a connection to the set of irrationals with continued fraction
digits exceeding a given number which tends to infinity.
