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A new lower bound for $$\|(3/2)^k\|$$. (English) Zbl 1127.11049
In 1957, Mahler proved that the inequality $$\| (p/q)^k\| <C^k,$$ where $$p>q>1$$ are relatively prime integers, has only finitely many solutions in positive integers $$k$$ for any $$C$$ satisfying $$0<C<1.$$ His proof is ineffective and gives no bound for $$k.$$ For applications to Waring’s problem, the lower bound $$\| (3/2)^k\| >0.75^k$$ for $$k \geq 5$$ would give an exact expression for the least integer $$g(k)$$ such that every positive integer can be expressed by a sum of at most $$k$$ positive powers. Thanks to some earlier papers of Baker and Coates, Beukers, the reviewer and, finally, Habsieger, the best known effective estimate up to now was $$\| (3/2)^k\| >0.577^k$$ for each $$k \geq K_0.$$
In this paper, the author improves this inequality to $$\| (3/2)^k\| >0.5803^k$$ for each $$k \geq K,$$ where $$K$$ is an effective constant. Although the improvement is numerically quite small, the proof is based on constructing a somewhat different sequence of approximations. It involves some very precise computations and, in addition, a careful selection of parameters. In conclusion, using the same method the author presents several sequences of approximations implying the estimates $$\| (4/3)^k\| > 0.4914^k$$ for $$k \geq K_1$$ and $$\| (5/4)^k\| > 0.5152^k$$ for $$k \geq K_2$$ improving (at least asymptotically) upon previous results of Bennett.

##### MSC:
 11J71 Distribution modulo one
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##### References:
 [1] A. Baker, J. Coates, Fractional parts of powers of rationals. Math. Proc. Cambridge Philos. Soc. 77 (1975), 269-279. · Zbl 0298.10018 [2] M. A. Bennett, Fractional parts of powers of rational numbers. Math. Proc. Cambridge Philos. Soc. 114 (1993), 191-201. · Zbl 0791.11030 [3] M. A. Bennett, An ideal Waring problem with restricted summands. Acta Arith. 66 (1994), 125-132. · Zbl 0793.11026 [4] F. Beukers, Fractional parts of powers of rationals. Math. Proc. Cambridge Philos. Soc. 90 (1981), 13-20. · Zbl 0466.10030 [5] G. V. Chudnovsky, Padé approximations to the generalized hypergeometric functions. I. J. Math. Pures Appl. (9) 58 (1979), 445-476. · Zbl 0434.10023 [6] F. Delmer, J.-M. Deshouillers, The computation of $$g(k)$$ in Waring’s problem. Math. Comp. 54 (1990), 885-893. · Zbl 0701.11043 [7] A. K. Dubickas, A lower bound for the quantity $$\Vert (3/2)^k\Vert$$. Russian Math. Surveys 45 (1990), 163-164. · Zbl 0712.11037 [8] L. Habsieger, Explicit lower bounds for $$\Vert (3/2)^k\Vert$$. Acta Arith. 106 (2003), 299-309. · Zbl 1126.11325 [9] J. Kubina, M. Wunderlich, Extending Waring’s conjecture up to $$471600000$$. Math. Comp. 55 (1990), 815-820. · Zbl 0725.11051 [10] K. Mahler, On the fractional parts of powers of real numbers. Mathematika 4 (1957), 122-124. · Zbl 0208.31002 [11] L. J. Slater, Generalized hypergeometric functions. Cambridge University Press, 1966. · Zbl 0135.28101 [12] R. C. Vaughan, The Hardy-Littlewood method. Cambridge Tracts in Mathematics 125, Cambridge University Press, 1997. · Zbl 0868.11046 [13] W. Zudilin, Ramanujan-type formulae and irrationality measures of certain multiples of $$π$$. Mat. Sb. 196:7 (2005), 51-66. · Zbl 1114.11064
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