Abstract
Let \({\mathbb {N}}\) be the set of all nonnegative integers. For a given set \(S\subset {\mathbb {N}}\) the representation function \(R_S(n)\) is defined as the number of solutions of the equation \(n=s+s'\), \(s<s'\), \(s,s'\in S\). Let l be a positive integer. In this paper, we prove that there exist sets A and B such that \({\mathbb {N}}=A\cup B\) and \(A\cap B=(2^{2l}-1)+(2^{2l+1}-1){\mathbb {N}}\) and \(R_A(n)=R_B(n)\ge 1\) for every positive integer n except for finite exceptions. This result improves the result of Chen and Lev.
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This work was supported by National Natural Science Foundation of China, Grant No. 11471017.
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Tang, M., Li, JW. Note on a result of Chen and Lev. Period Math Hung 79, 134–140 (2019). https://doi.org/10.1007/s10998-019-00286-1
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DOI: https://doi.org/10.1007/s10998-019-00286-1