On complete sequences

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• 作者：J.-H. Fang ; X.-Y. Liu
• 关键词：complete sequence ; Hegyvári’s Theorem ; Lebesgue measure ; 11B13 ; 11B75
• 刊名：Acta Mathematica Hungarica
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：148
• 期：1
• 页码：211-221
• 全文大小：536 KB
• 参考文献：1.Birch B. J.: Note on a problem of Erdős, Proc. Cambridge Philos. Soc. 55, 370–373 (1959)MathSciNet CrossRef MATH
  2.Burr S. A., Erdős P., Graham R. L., Wen-Ching Li W.: Complete sequences of sets of integer powers. Acta Arith. 77, 133–138 (1996)MathSciNet
  3.Chen Y.-G., Fang J.-H.: Remark on the completeness of an exponential type sequence. Acta Math. Hungar. 136, 189–195 (2012)MathSciNet CrossRef MATH
  4.Chen Y.-G., Fang J.-H.: Hegyvári’s theorem on complete sequences. J. Number Theory, 133, 2857–2862 (2013)MathSciNet CrossRef MATH
  5.Graham R. L.: Complete sequences of polynomial values. Duke Math. J. 31, 275–285 (1964)MathSciNet CrossRef MATH
  6.Hegyvári N.: Additive properties of sequences of multiplicatively perturbed square values. J. Number Theory, 54, 248–260 (1995)MathSciNet CrossRef MATH
  7.Hegyvári N.: Complete sequences in N 2. European J. Combin. 17, 741–749 (1996)MathSciNet CrossRef MATH
  8.Hegyvári N.: On the completeness of an exponential type sequence. Acta Math. Hungar. 86, 127–135 (2000)MathSciNet CrossRef
  
• 作者单位：J.-H. Fang (1)
  X.-Y. Liu (1)
  
  1. Department of Mathematics, Nanjing University of Information Science & Technology, Nanjing, 210044, P.R. China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Sciences
  Mathematics
  
• 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
• ISSN：1588-2632

文摘

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence \({S=\{s_{1}, s_{2}, \dots\}}\) of positive integers and a positive real number α, let S _α denote the sequence \({\{\lfloor\alpha s_{1}\rfloor, \lfloor\alpha s_{2}\rfloor, \dots\}}\), where \({\lfloor x \rfloor}\) denotes the greatest integer not greater than x. Let \({{U_S = \{\alpha \mid S_\alpha} \, is complete\}}\). Hegyvári [6] proved that if \({\lim_{n\to\infty} (s_{n+1}-s_{n})=+ \infty}\), \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma < 2}\), and \({U_{S}\ne\emptyset}\), then \({\mu(U_{S}) > 0}\), where \({\mu(U_{S})}\) is the Lebesgue measure of U _S . Yong-Gao Chen and the first author [4] proved that, if \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma \leqq 7/4=1.75}\), then \({\mu(U_{S}) > 0}\). In this paper, we prove that the conclusion holds for \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\). Mathematics Subject Classification 11B13 11B75