The arithmetic-geometric scaling spectrum for continued fractions

Abstract

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.