Additive Complements with Narkiewicz's Condition

Abstract

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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References

  1. [1]

    Y. G. Chen and J. H. Fang: On additive complements, II, Proc. Amer. Math. Soc. 139 (2011), 881–883.

    MathSciNet  Article  Google Scholar 

  2. [2]

    Y. G. Chen and J. H. Fang: On a conjecture of Sárközy and Szemerédi, Acta Arith. 169 (2015), 47–58.

    MathSciNet  Article  Google Scholar 

  3. [3]

    Y. G. Chen and J. H. Fang: Additive complements of the squares, J. Number Theory 180 (2017), 410–422.

    MathSciNet  Article  Google Scholar 

  4. [4]

    L. Danzer: Über eine Frage von G. Hanani aus der additiven Zahlentheorie, J. Reine Angew. Math. 214/215 (1964), 392–394.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    P. Erdős: Some unsolved problems, Mich. J. Math. 4 (1957), 291–300.

    MathSciNet  Article  Google Scholar 

  6. [6]

    P. Erdős: Some unsolved problems, Publ. Math. Inst. Hung. Acad. Sci., Ser A 6 (1961), 221–254.

    MathSciNet  MATH  Google Scholar 

  7. [7]

    P. Erdős and R. L. Graham: Old and New Problems and Results in Combinatorial Number Theory, Monographies de L'Enseignement Mathématique, 28, Université de Genéve, Geneva, (1980).

    Google Scholar 

  8. [8]

    J. H. Fang and Y. G. Chen: On additive complements, Proc. Amer. Math. Soc. 138 (2010), 1923–1927.

    MathSciNet  Article  Google Scholar 

  9. [9]

    J. H. Fang and Y. G. Chen: On finite additive complements, Discrete Math. 313 (2013), 595–598.

    MathSciNet  Article  Google Scholar 

  10. [10]

    J. H. Fang and Y. G. Chen: On additive complements. III, J. Number Theory 141 (2014), 83–91.

    MathSciNet  Article  Google Scholar 

  11. [11]

    J. H. Fang and Y. G. Chen: On infinite additive complements, Sci. China Math. 60 (2017), 1779–1790.

    MathSciNet  Article  Google Scholar 

  12. [12]

    H. Halberstam and K. F. Roth: Sequences, 2nd ed., Springer-Verlag, New York-Berlin 1983.

    Google Scholar 

  13. [13]

    S. Z. Kiss, E. Rozgonyi and C. Sándor: On additive complement of a finite set, J. Number Theory 136 (2014), 195–203.

    MathSciNet  Article  Google Scholar 

  14. [14]

    W. Narkiewicz: Remarks on a conjecture of Hanani in additive number theory, Colloq. Math. 7 (1959/60), 161–165.

    MathSciNet  Article  Google Scholar 

  15. [15]

    I. Z. Ruzsa: Additive completion of lacunary sequences, Combinatorica 21 (2001), 279–291.

    MathSciNet  Article  Google Scholar 

  16. [16]

    I. Z. Ruzsa: Exact additive complements, Quart. J. Math. 68 (2017), 227–235.

    MathSciNet  MATH  Google Scholar 

  17. [17]

    A. Sárközy and E. Szemerédi: On a problem in additive number theory, Acta Math. Hungar. 64 (1994), 237–245.

    MathSciNet  Article  Google Scholar 

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Correspondence to Yong-Gao Chen.

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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). https://doi.org/10.1007/s00493-018-3947-2

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Mathematics Subject Classification (2010)

  • 11B13
  • 11B34