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

  • Kurt Mahler


Let denote the distance of the real number to the nearest integer. In this paper, Mahler proves that, if and are coprime integers satisfying and 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 . He uses this result to show that, except for a finite number of values ,

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

where is the function in Waring's problem.

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