A UNIQUE REPRESENTATION BI-BASIS FOR THE INTEGERS. II
2016 ◽
Vol 94
(1)
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pp. 1-6
Keyword(s):
For$n\in \mathbb{Z}$and$A\subseteq \mathbb{Z}$, define$r_{A}(n)$and${\it\delta}_{A}(n)$by$r_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}+a_{2},a_{1}\leq a_{2}\}$and${\it\delta}_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}-a_{2}\}$. We call$A$a unique representation bi-basis if$r_{A}(n)=1$for all$n\in \mathbb{Z}$and${\it\delta}_{A}(n)=1$for all$n\in \mathbb{Z}\setminus \{0\}$. In this paper, we prove that there exists a unique representation bi-basis$A$such that$\limsup _{x\rightarrow \infty }A(-x,x)/\sqrt{x}\geq 1/\sqrt{2}$.
2014 ◽
Vol 7
(3)
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Keyword(s):
1992 ◽
Vol 20
(2)
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pp. 1143-1145
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1961 ◽
Vol 13
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pp. 557-568
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2018 ◽
Vol 29
(09)
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pp. 1850082
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2010 ◽
Vol 22
(6)
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pp. 767-776
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2007 ◽
Vol 28
(1)
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pp. 33-35
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