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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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