Rational approximations of the exponential function at rational points

TL;DR

This paper develops explicit and asymptotic lower bounds for approximating exponential functions at rational points using generalized continued fractions, improving previous results for certain parameter ranges.

Contribution

It introduces new bounds for approximations of e^{s/t} and enhances existing results by extracting common factors in continued fraction convergents for |s| ≥ 3.

Findings

Established explicit lower bounds for |e^{s/t} - M/N|

Improved asymptotic bounds for large |s|

Enhanced approximation techniques using generalized continued fractions

Abstract

We give explicit and asymptotic lower bounds for the quantity $|e^{s/t}-M/N|$ by studying a generalized continued fraction expansion of $e^{s/t}$. In cases $|s|\geq 3$ we improve existing results by extracting a large common factor from the numerators and the denominators of the convergents of that generalized continued fraction.