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 $P(\xi )\ne 0.$ Let $k\ge 1$ and $\omega (k,H)$ be the infimum of the numbers $r>0$ satisfying the estimate $$\left|{\lambda }_0+{\lambda }_1\xi +{\lambda }_2{\xi }^2+\dots +{\lambda }_k{\xi }^k\right|>\frac{1}{H^r},$$ for all $({\lambda }_0,\dots ,{\lambda }_k{)}^T\in {\mathbb{Z}}^{k+1}\setminus \{\overline{0}\}$ 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.

This collaboration has come up due to the wonderful programme: Women in Numbers Europe 4