Unique representation bi-basis for the integers

Authors

  • R. Xiong
  • M. Tang

Keywords:

bi-basis, representation function

Abstract

For \(n\in\mathbb{Z}\) and \(A\subseteq\mathbb{Z},\) let \(r_{A}(n)=\# \{(a_{1}, a_{2})\in A^{2}: n=a_{1}+a_{2}, a_{1}\leq a_{2}\}\) and \(\delta_{A}(n)=\# \{(a_{1}, a_{2})\in A^{2}: n=a_{1}-a_{2} \}.\) We call \(A\) a unique representation bi-basis if \(r_{A}(n)=1\) for all \(n\in\mathbb{Z}\) and \(\delta_{A}(n)=1\) for all \(n\in\mathbb{Z}\setminus\{0\}.\) In this paper, we construct a unique representation bi-basis of \(\mathbb{Z}\) whose growth is logarithmic. DOI: 10.1017/S0004972713000762

Published

2014-03-25

Issue

Vol. 89 No. 3 (2014)

Section

Articles
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