Partitions of natural numbers with the same weighted representation functions

Yang, Quan-Hui; Chen, Yong-Gao

doi:10.1016/j.jnt.2012.06.005

Cited by 23 publications

(10 citation statements)

References 15 publications

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“…For related results, one may refer to [1][2][3][4] and [6][7][8][9][10][11]. In this paper, we prove further results about these representation functions.…”

Section: Introductionsupporting

confidence: 58%

Representation Functions on Abelian Groups

Chen

2018

Bull. Aust. Math. Soc.

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Let $G$ be a finite abelian group, $A$ a nonempty subset of $G$ and $h\geq 2$ an integer. For $g\in G$, let $R_{A,h}(g)$ denote the number of solutions of the equation $x_{1}+\cdots +x_{h}=g$ with $x_{i}\in A$ for $1\leq i\leq h$. Kiss et al. [‘Groups, partitions and representation functions’, Publ. Math. Debrecen85(3) (2014), 425–433] proved that (a) if $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $|G|=2|A|$, and (b) if $h$ is even and $|G|=2|A|$, then $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$. We prove that $R_{G\setminus A,h}(g)-(-1)^{h}R_{A,h}(g)$ does not depend on $g$. In particular, if $h$ is even and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for some $g\in G$, then $|G|=2|A|$. If $h>1$ is odd and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $R_{A,h}(g)=\frac{1}{2}|A|^{h-1}$ for all $g\in G$. If $h>1$ is odd and $|G|$ is even, then there exists a subset $A$ of $G$ with $|A|=\frac{1}{2}|G|$ such that $R_{A,h}(g)\not =R_{G\setminus A,h}(g)$ for all $g\in G$.

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“…For related results, one may refer to [1][2][3][4] and [6][7][8][9][10][11]. In this paper, we prove further results about these representation functions.…”

Section: Introductionsupporting

confidence: 58%

Representation Functions on Abelian Groups

Chen

2018

Bull. Aust. Math. Soc.

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“…for all sufficiently large integers n. By using the Thue-Morse sequence, Dombi [6] answered Sárközy's problem affirmatively. In the last few decades, there are many results about representation functions (see, for example, [2,4,5,9,7,10,11,12,13,14,15,16,17]).…”

Section: Introductionmentioning

confidence: 99%

Partitions of finite nonnegative integer sets with identical representation functions

Sun¹,

Pan²

2022

Preprint

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“…Later Lev [7], Sándor [9], and Tang [10] gave a simple proof of this result respectively. For related results, one can refer to [4], [12]- [14].…”

Section: Introductionmentioning

confidence: 99%

On the integer sets with the same representation functions

JIAO¹,

Sándor²,

Yang³

et al. 2021

Preprint

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Let N be the set of all nonnegative integers. For S ⊆ N and n ∈ N, let RS(n) denote the number of solutions of the equation n = s1 + s2, s1, s2 ∈ S and s1 < s2. Let A be the set of all nonnegative integers which contain an even number of digits 1 in their binary representations andThis solved a problem of Chen and Lev. In this paper, we prove that, if C ∪ D = [0, m] \ {r} with 0 < r < m, C ∩ D = ∅ and 0 ∈ C, then RC(n) = RD(n) for any nonnegative integer n if and only if there exists an integer l ≥ 2 such that m = 2 l , r = 2 l−

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Partitions of natural numbers with the same weighted representation functions

Cited by 23 publications

References 15 publications

Representation Functions on Abelian Groups

Representation Functions on Abelian Groups

Partitions of finite nonnegative integer sets with identical representation functions

On the integer sets with the same representation functions

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