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

  • Kurt Mahler


Let x\|x\| denote the distance of the real number xx to the nearest integer. In this paper, Mahler proves that, if uu and vv are coprime integers satisfying u>v2u>v\ge 2 and ε>0\varepsilon>0 is an arbitrarily small positive number, the inequality

\par\par(uv)n<eεn\par\par\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 nn. He uses this result to show that, except for a finite number of values kk,

\par\parg(k)=2k(32)k2,\par\par\par \par g(k)=2^k-\left\lfloor\left(\frac{3}{2}\right)^k\right\rfloor-2, \par \par

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

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