The Minkowski $?(x)$ function, a class of singular measures, quasi-modular and mean-modular forms. I

TL;DR

This paper links the Minkowski question mark function to mean-modular forms, establishing a framework that connects it to modular objects and quasi-modular forms, revealing new arithmetic and analytical insights.

Contribution

It introduces mean-modular forms as a new class of functions generalizing modular forms, and constructs an isomorphism with quasi-modular forms, providing a novel interpretation of the Minkowski question mark function.

Findings

Established a link between ?(x) and mean-modular forms.

Constructed a class of measures related to modular forms.

Proved an isomorphism between quasi-modular and mean-modular forms.

Abstract

The Minkowski question mark function is a rich object which can be explored from the perspective of dynamical systems, complex dynamics, metric number theory, multifractal analysis, transfer operators, integral transforms, and as a function itself via analysis of continued fractions and convergents. Our permanent target, however, was to get arithmetic interpretation of the moments of ?(x) (which are relatives of periods of Maass wave forms) and to relate the function ?(x) to certain modular objects. In this paper we establish this link, embedding ?(x) not into the modular-world itself, but into a space of functions which are generalizations and which we call mean-modular forms. For this purpose we construct a wide class of measures, and also investigate modular forms for congruence subgroups which additionally satisfy the three term functional equation. From this perspective, the modular forms for the whole modular group as well as the Stieltjes transform of ?(x) (the dyadic period function) minus the Eisenstein series of weight 2 fall under the same uniform definition. The main result is the construction of the canonical isomorphism between the spaces of quasi-modular forms and mean-modular forms. This gives unexpected Minkowski question mark function-related interpretation of quasi-modular forms.