Extended-cycle integrals of modular functions for badly approximable numbers

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. Contents 1. Introduction 1 Acknowledgement 5 2. Fundamental properties of the extended val function 5 3. The extended val function as the limit value along a geodesic 5 4. The extended val function as the limit value along a continued fraction 9 5. Elementary units for badly approximable numbers 13 6. An explicit computation of values of extended val function 14 7. The value of val function at the Thue-Morse word 16 References 17