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UNIQUE REPRESENTATION BI-BASIS FOR THE INTEGERS

  • RAN XIONG (a1) and MIN TANG (a1)

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.

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References

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[1]Chen, Y. G., ‘The difference basis and bi-basis of ${ \mathbb{Z} }_{m} $’, J. Number Theory 130 (2010), 716726.
[2]Cilleruelo, J. and Nathanson, M. B., ‘Perfect difference sets constructed from Sidon sets’, Combinatorica 28 (2008), 401414.
[3]Lee, J., ‘Infinitely often dense bases for the integers with a prescribed representation function’, Integers 10 (2010), 299307.
[4]Nathanson, M. B., ‘Unique representation bases for integers’, Acta Arith. 108 (2003), 18.
[5]Tang, M., ‘Dense sets of integers with a prescribed representation function’, Bull. Aust. Math. Soc. 84 (2011), 4043.
[6]Tang, C. W., Tang, M. and Wu, L., ‘Unique difference bases of $ \mathbb{Z} $’, J. Integer Seq. 14 (2011), Article 11.1.8.
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