# Generalizations of some results about the regularity properties of an additive representation function

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## Abstract

Let \({A = \{a_{1},a_{2},\dots{}
\}}\) \({(a_{1} < a_{2} < \cdots )}\) be an infinite sequence of nonnegative integers, and let \({R_{A,2}(n)}\) denote the number of solutions of \({a_{x}+a_{y}=n}\) \({(a_{x},a_{y} \in A)}\). P. Erdős, A. Sárközy and V. T. Sós proved that if \({\lim_{N\to\infty}\frac{B(A,N)}{\sqrt{N}}=+\infty}\) then \({|\Delta_{1}(R_{A,2}(n))|}\) cannot be bounded, where \({B(A,N)}\) denotes the number of blocks formed by consecutive integers in *A* up to *N* and \({\Delta_{l}}\) denotes the *l*-th difference. Their result was extended to \({\Delta_{l}(R_{A,2}(n))}\) for any fixed \({l\ge2}\). In this paper we give further generalizations of this problem.

## Key words and phrases

additive number theory general sequence additive representation function## Mathematics Subject Classification

primary 11B34## Preview

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## References

- 1.N. Alon and J. Spencer,
*The Probabilistic Method*, 4th Ed., Wiley (New York, 2016).Google Scholar - 2.Erdős P., Sárközy A.: Problems and results on additive properties of general sequences. I. Pacific J. Math.,
**118**, 347–357 (1985)MathSciNetCrossRefGoogle Scholar - 3.Erdős P., Sárközy A.: Problems and results on additive properties of general sequences. II. Acta Math. Hungar.,
**48**, 201–211 (1986)MathSciNetCrossRefGoogle Scholar - 4.Erdős P., Sárközy A., Sós V. T.: Problems and results on additive properties of general sequences. III. Studia Sci. Math. Hungar.,
**22**, 53–63 (1987)MathSciNetzbMATHGoogle Scholar - 5.H. Halberstam and K. F. Roth,
*Sequences*, Springer-Verlag (New York, 1983).Google Scholar - 6.Kiss S.: Generalization of a theorem on additive representation functions. Ann. Univ. Sci. Budapest. Eötvös Sect. Math.,
**48**, 15–18 (2005)MathSciNetzbMATHGoogle Scholar - 7.Kiss S.: On a regularity property of additive representation functions. Periodica Math. Hungar.,
**51**, 31–35 (2005)MathSciNetCrossRefGoogle Scholar - 8.Kiss S. Z.: On the
*k*-th difference of an additive representation function. Studia Sci. Math. Hungar.,**48**, 93–103 (2011)MathSciNetzbMATHGoogle Scholar - 9.Sárközy A.: On additive representation functions of finite sets. I (Variation). Periodica Math. Hungar.,
**66**, 201–210 (2013)MathSciNetCrossRefGoogle Scholar

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