Extended-cycle integrals of modular functions for badly approximable numbers

TL;DR

This paper extends the concept of cycle integrals of modular functions from real quadratic numbers to badly approximable numbers and provides explicit representations for some cases, advancing understanding in number theory.

Contribution

It introduces a new extension of cycle integrals to badly approximable numbers and offers explicit formulas for specific instances, broadening the scope of modular function analysis.

Findings

Extended cycle integrals to badly approximable numbers.

Explicit representations for certain cases of extended-cycle integrals.

Enhanced understanding of modular functions in number theory.

Abstract

Cycle integrals of modular functions are expected to play a role in real quadratic analogue of singular moduli. In this paper, we extend the definition of cycle integrals of modular functions from real quadratic numbers to badly approximable numbers. We also give explicit representations of values of extended-cycle integrals for some cases.