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On a lower bound for \(||(4/3)^{k}||\). (English) Zbl 1326.11034
Let \(||x||\) be the distance from \(x\) to nearest integer. In a previous paper [Math. Notes 85, No. 5–6, 877–885 (2009; Zbl 1230.11088)], the author proved \(||(4/3)^k||>0.4910^k\) for \(k\geq5868122745713241570\). In this paper, he proves that \(||(4/3)^k||>(4/9)^k\) for \(k\geq 6\). He uses the same method as W. Zudilin [J. Théor. Nombres Bordx. 19, No. 1, 311–323 (2007; Zbl 1127.11049)] for \(k\geq 17545718\) and then for remaining \(k\) he used software of F. Delmer and J.-M. Deshouillers [Math. Comput. 54, No. 190, 885–893 (1990; Zbl 0701.11043)]. Using a result of M. A. Bennett [Acta Arith. 66, No. 2, 125–132 (1994; Zbl 0793.11026)], the author obtains that the order of the set \(\{1^k,3^k,4^k,5^k,\dots\}\) as an additive basis for the positive integers is \(3^k+[(4/3)^k]-2\).
11J54 Small fractional parts of polynomials and generalizations
11J25 Diophantine inequalities
11J04 Homogeneous approximation to one number
11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
Full Text: DOI arXiv
[1] Bennett M. A., Acta Arith. 66 pp 125– (1994)
[2] Delmer F., Math. Comp. 54 pp 885– (1990)
[3] Fikhtengolts G. M., The Course of Differential and Integral Calculus 2 (1966)
[4] DOI: 10.1134/S0001434609050289 · Zbl 1230.11088 · doi:10.1134/S0001434609050289
[5] Rosser J. B., Math. Comp. 29 pp 243– (1975)
[6] DOI: 10.5802/jtnb.588 · Zbl 1127.11049 · doi:10.5802/jtnb.588
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