## 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

### Keywords:

fractional parts; powers of rationals

### Citations:

Zbl 0208.31002; Zbl 0261.10025
Full Text:

### 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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