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ON MONOTONE INCREASING REPRESENTATION FUNCTIONS

Part of: Sequences and sets

Published online by Cambridge University Press:  17 July 2023

SÁNDOR Z. KISS Open the ORCID record for SÁNDOR Z. KISS [Opens in a new window] ,
CSABA SÁNDOR Open the ORCID record for CSABA SÁNDOR [Opens in a new window]  and
QUAN-HUI YANG Open the ORCID record for QUAN-HUI YANG [Opens in a new window]
SÁNDOR Z. KISS*
Affiliation:
Department of Algebra and Geometry, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
CSABA SÁNDOR
Affiliation:
Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary; Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary; MTA-BME Lendület Arithmetic Combinatorics Research Group, ELKH, Műegyetem rkp. 3., H-1111 Budapest, Hungary e-mail: csandor@math.bme.hu
QUAN-HUI YANG
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, PR China e-mail: yangquanhui01@163.com
*
e-mail: ksandor@math.bme.hu
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Abstract

Let $k\ge 2$ be an integer and let A be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set A is the number of representations of a nonnegative integer n as the sum of k terms from A. Let $A(n)$ denote the counting function of A. Bell and Shallit [‘Counterexamples to a conjecture of Dombi in additive number theory’, Acta Math. Hung., to appear] recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{{(k-2)}/{k}-\epsilon })$ for some $\epsilon>0$, then $R_{\mathbb {N}\setminus A,k}(n)$ is eventually strictly increasing. We improve this result to $A(n)=O(n^{{(k-2)}/{(k-1)}})$. We also give an example to show that this bound is best possible.

Keywords

additive number theoryadditive representation functionmonotonicity

MSC classification

Primary: 11B34: Representation functions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 2 , April 2024 , pp. 196 - 205
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by the NKFIH Grant No. K129335; the second author was supported by the NKFIH Grant No. K129335.

References

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