Reprint: On the fractional parts of the powers of a rational number. II (1957)

Abstract

Let $\|x\|$ denote the distance of the real number $x$ to the nearest integer. In this paper, Mahler proves that, if $u$ and $v$ are coprime integers satisfying $u>v\ge 2$ and $\varepsilon>0$ is an arbitrarily small positive number, the inequality

$\par \par \left\|\left(\frac{u}{v}\right)^n\right\|<e^{\varepsilon n} \par \par$

is satisfied by at most a finite number of positive integer solutions $n$. He uses this result to show that, except for a finite number of values $k$,

$\par \par g(k)=2^k-\left\lfloor\left(\frac{3}{2}\right)^k\right\rfloor-2, \par \par$

where $g(k)$ is the function in Waring's problem.

Reprint of the author's paper [Mathematika 4, 122--124 (1957; Zbl 0208.31002)].