A Diophantine Approximation of $e^{1/s}$ in Terms of Integrals

Abstract

Let $p_n/q_n$ be the $n$-th convergent of the continued fraction expansion of a real number $\alpha$. It is known that $|p_n-q_n\alpha|$ is very small tending to $0$ as $n$ tends to infinity. In this paper we establish a method how to express $p_n-q_n\alpha$ in terms of integrals when $\alpha$ is an $e$-type real number and its continued fraction expansion is quasi-periodic.