Additive Complements with Narkiewicz's Condition


Two sequences A and B of non-negative integers are called additive complements, if their sum contains all suffciently large integers. Let A(x) and B(x) be the counting functions of A and B, respectively. In 1994, Sárközy and Szemerédi proved that, for additive complements A and B, if limsup A(x)B(x)=x ≤ 1, then A(x)B(x)-x→+∞ as x→+∞. In this paper, motivated by a recent result of Ruzsa, we prove the following result: for additive complements A, B with Narkiewicz's condition: A(2x)=A(x)→ as x→+∞, we have A(x)B(x)-x>(1+o(1))a(x)=A(x) as x→+∞, where a(x) is the largest element in A⋂[1,x]. Furthermore, this is the best possible. As a corollary, for additive complements A, B with Narkiewicz's condition: A(2x)=A(x)→1 as x→+∞ and any M >1, we have A(x)B(x)-x>A(x)M for all suffciently large x.

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Corresponding author

Correspondence to Yong-Gao Chen.

Additional information

The authors are supported by the National Natural Science Foundation of China, Grant Nos. 11771211, 11671211.

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Chen, YG., Fang, JH. Additive Complements with Narkiewicz's Condition. Combinatorica 39, 813–823 (2019).

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Mathematics Subject Classification (2010)

  • 11B13
  • 11B34