Mahler, Kurt

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

Doc. Math. Extra Vol. Mahler Selecta, 595-598 (2019)
DOI: 10.25537/dm.2019.SB-595-598


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 \[ \par \par \left\|\left(\frac{u}{v}\right)^n\right\|<e^{\varepsilon n} \par \par \] \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 \[ \par \par g(k)=2^k-\left\lfloor\left(\frac{3}{2}\right)^k\right\rfloor-2, \par \par \] \par where \(g(k)\) is the function in Waring's problem. \par Reprint of the author's paper [Mathematika 4, 122--124 (1957; Zbl 0208.31002)].

Mathematics Subject Classification

11-03, 11J25