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.
Similar content being viewed by others
References
Y. G. Chen and J. H. Fang: On additive complements, II, Proc. Amer. Math. Soc. 139 (2011), 881–883.
Y. G. Chen and J. H. Fang: On a conjecture of Sárközy and Szemerédi, Acta Arith. 169 (2015), 47–58.
Y. G. Chen and J. H. Fang: Additive complements of the squares, J. Number Theory 180 (2017), 410–422.
L. Danzer: Über eine Frage von G. Hanani aus der additiven Zahlentheorie, J. Reine Angew. Math. 214/215 (1964), 392–394.
P. Erdős: Some unsolved problems, Mich. J. Math. 4 (1957), 291–300.
P. Erdős: Some unsolved problems, Publ. Math. Inst. Hung. Acad. Sci., Ser A 6 (1961), 221–254.
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).
J. H. Fang and Y. G. Chen: On additive complements, Proc. Amer. Math. Soc. 138 (2010), 1923–1927.
J. H. Fang and Y. G. Chen: On finite additive complements, Discrete Math. 313 (2013), 595–598.
J. H. Fang and Y. G. Chen: On additive complements. III, J. Number Theory 141 (2014), 83–91.
J. H. Fang and Y. G. Chen: On infinite additive complements, Sci. China Math. 60 (2017), 1779–1790.
H. Halberstam and K. F. Roth: Sequences, 2nd ed., Springer-Verlag, New York-Berlin 1983.
S. Z. Kiss, E. Rozgonyi and C. Sándor: On additive complement of a finite set, J. Number Theory 136 (2014), 195–203.
W. Narkiewicz: Remarks on a conjecture of Hanani in additive number theory, Colloq. Math. 7 (1959/60), 161–165.
I. Z. Ruzsa: Additive completion of lacunary sequences, Combinatorica 21 (2001), 279–291.
I. Z. Ruzsa: Exact additive complements, Quart. J. Math. 68 (2017), 227–235.
A. Sárközy and E. Szemerédi: On a problem in additive number theory, Acta Math. Hungar. 64 (1994), 237–245.
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors are supported by the National Natural Science Foundation of China, Grant Nos. 11771211, 11671211.