The arithmetic-geometric scaling spectrum for continued fractions

TL;DR

This paper introduces the arithmetic-geometric scaling spectrum for continued fractions, fully characterizes its multifractal spectrum using a free energy function, and explores its geometric and number-theoretic properties.

Contribution

It provides a complete determination of the multifractal spectrum for the new scaling, linking it to a free energy function and Hausdorff dimension analysis.

Findings

Hausdorff dimension matches Legendre transform of free energy function

Complete characterization of the multifractal spectrum

Asymptotic behavior at the spectrum's boundary identified

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.