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.
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Acknowledgements
The author would like to show my greatest appreciation to Professor Takuya Yamauchi for giving me many pieces of advice. Author deeply grateful to Dr. Toshiki Matsusaka for giving many comments. Author would like to express my gratitude to Professor Shun’ichi Yokoyama and Dr. Toshihiro Suzuki for giving me constructive comments regarding a failed attempt to compute \( {{\,\textrm{val}\,}}(x) \) numerically. It is a pleasure to extend my thanks to Professor Tatsuya Tate for teaching me ergodic theory. Author also thank Dr. Daisuke Kazukawa, Dr. Hiroki Nakajima, and Dr. Shin’ichiro Kobayashi for teaching me geodesics, hyperbolic geometry, and metric spaces. Author appreciate the technical assistance of Dr. Naruaki Kato for introducing me to how to write works by using GitHub. Author thank the referees for their helpful suggestions and comments which substantially improved the presentation of our paper.
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The author is supported by JSPS KAKENHI Grant Number JP 20J20308.
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Murakami, Y. Extended-cycle integrals of modular functions for badly approximable numbers. Res. number theory 9, 50 (2023). https://doi.org/10.1007/s40993-023-00457-7
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DOI: https://doi.org/10.1007/s40993-023-00457-7