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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 $$3^k+[(4/3)^k]-2$$.
##### MSC:
 11J54 Small fractional parts of polynomials and generalizations 11J25 Diophantine inequalities 11J04 Homogeneous approximation to one number 11B57 Farey sequences; the sequences $$1^k, 2^k, \dots$$
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##### References:
 [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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