On the multiplicative group generated by \({\{ \frac{[\sqrt{2}n]}{n} | n \in \mathbb{N} \}}\

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• 作者：I. Kátai ; B. M. Phong
• 关键词：completely additive functions ; multiplicative group ; 11K65 ; 11N37 ; 11N64
• 刊名：Acta Mathematica Hungarica
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：145
• 期：1
• 页码：80-87
• 全文大小：261 KB
• 参考文献：1. Hildebrand A.: An Erd?s–Wintner theorem for differences of additive functions, / Trans. Amer. Math. Soc., 310, 257-76 (1988)
  
• 作者单位：I. Kátai (1)
  B. M. Phong (1)
  
  1. Department of Computer Algebra, Faculty of Informatics, E?tv?s Loránd University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary
  
• 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

We prove that the multiplicative group generated by \({\{ \frac{[\sqrt{2}n]}{n} | n \in \mathbb{N} \}}\) is the group of positive rational numbers. It is proved that if a completely additive function f satisfying f \({([\sqrt{2}n]) - f(n) \rightarrow C (n \rightarrow \infty)}\) for some real number C, then \({f(n) = A{\text{log}} n}\) , where \({A = \frac{2C}{log 2}}\) .