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Fractional parts of powers of rationals. (English) Zbl 0298.10018

Let \(a\) and \(q\) be any relatively prime integers with \(a >q\ge 2\) Mahler proved in 1957 that for any \(\varepsilon >0\) there exist only finitely many positive integers \(n\) such that \(\vert(a/q)^n\Vert < e^{-\varepsilon n}\); here \(\Vert x\Vert\) denotes the distance of \(x\) from the nearest integer [see K. Mahler, Mathematika 4, 122–124 (1957; Zbl 0208.31002)]. The authors establish an effective theorem which is slightly weaker: There exist effectively computable numbers \(N\) and \(\eta\), with \(0< \eta <1\), such that \(\vert(a/q)\Vert^n > q^{-\eta n}\) for all integers \(n\ge N\). It is not possible from this result to derive that the number \(g(k)\) in Waring’s problem is given by \[ 2^k + \left[\left(\frac32\right)^k\right] - 2 \] for sufficiently large \(k\), as follows from Mahler’s result. The theorem in the present paper is based on the following \(p\)-adic analogue of Baker’s result in [Acta Arith. 24, 33–36 (1973; Zbl 0261.10025)]. Let \(p\) be a prime and \(a\) any non-zero integer. There is an effectively computable number \(c=c(a,p)\) such that, for any \(\delta\) with \(0< \delta \le \tfrac12\), the inequalities \[ 0 < \vert a^n - b\vert_p < \delta^{c\log \vert 2b\vert} e^{-\delta n} \] have no solution in integers \(b\) and \(n>0\). In the proof a new device has been used to overcome the fact that the \(p\)-adic exponential function converges only in a certain neighbourhood of the origin.

MSC:

11J54 Small fractional parts of polynomials and generalizations
11J61 Approximation in non-Archimedean valuations
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References:

[1] Mahler, Mathematika 4 pp 122– (1957)
[2] Coates, Acta Arith. 15 pp 279– (1969)
[3] Baker, Mathematika 13 pp 204– (1966)
[4] Baker, Acta Arith. 21 pp 117– (1972)
[5] Baker, Acta Arith. 24 pp 33– (1973)
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