×

zbMATH — the first resource for mathematics

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 \(1^k, 2^k, \dots\)
PDF BibTeX XML Cite
Full Text: DOI arXiv
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.