Transcendence measure of $e^{1/n}$

Abstract

For a given transcendental number $\xi$ and for any polynomial $P(X)=: \lambda_0+\cdots+\lambda_k X^k \in \mathbb{Z}[X]$, we know that $ P(\xi) \neq 0.$ Let $k \geq 1$ and $\omega (k, H)$ be the infimum of the numbers $r > 0$ satisfying the estimate $$ \left|\lambda_0+\lambda_1 \xi+\lambda_2 \xi^{2}+ \ldots +\lambda_k\xi^{k}\right| > \frac{1}{H^r}, $$ for all $(\lambda_0, \ldots ,\lambda_k)^T \in \mathbb{Z}^{k+1}\setminus\{\overline{0}\}$ with $\max_{1\le i\le k} \{|\lambda_i|\} \le H$. Any function greater than or equal to $\omega (k, H)$ is a {\it transcendence measure of $\xi$}. In this article, we find out a transcendence measure of $ e^{1/n}$ which improves a result proved by Mahler(\cite{Mahler}) in 1975.