# On a Problem of Partitions of ℤ 𝑚 with Identical Representation Functions

## Main Article Content

## Abstract

For every positive integer 𝑚, ℤ𝑚 represents the set of residue class modulo 𝑚 such that ℤ 𝑚 = {0̄ , 1̄ , 2̄ , . . . , 𝑚 - 1} when 𝑥̄ = {𝑦 ∈ ℤ: 𝑥 ≡ 𝑦(mod 𝑚)}. For 𝐴 ⊆ ℤ𝑚 and 𝑛̄ ∈ ℤ 𝑚, the representation function was defined as

𝑅1(𝐴, 𝑛̄ ) = |{(𝑎̄, 𝑏̄ ) ∈ 𝐴 × 𝐴: 𝑎̄ ⊕ 𝑏̄ = 𝑛̄ }|.

In this paper, we studied Sárközy’s problem in ℤ𝑚 where the two subsets 𝐴 and 𝐵 of ℤ 𝑚 caused |(𝐴 ∪ 𝐵)\(𝐴 ∩ 𝐵)| = 𝑚 - 3 and 𝑅1(𝐴, 𝑛̄ ) = 𝑅1(𝐵, 𝑛̄ ) for all 𝑛̄ ∈ ℤ𝑚.

## Article Details

*PKRU SciTech Journal*,

*8*(1), 44–53. Retrieved from https://ph01.tci-thaijo.org/index.php/pkruscitech/article/view/252470

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

- The original content that appears in this journal is the responsibility of the author excluding any typographical errors.
- The copyright of manuscripts that published in PKRU SciTech Journal is owned by PKRU SciTech Journal.

## References

Nathanson, M. (1978). Representation functions of sequences in additive number

theory (pp. 16 - 20). In 72th Proceedings of the American Mathematical Society (PROC),

USA.

Dombi, G. (2002). Additive properties of certain sets. Acta Arithmetica, 103, 137–146.

Chen, Y. G., & Wang, B. (2003). On additive properties of two special sequences. Acta

Arithmetica, 110, 299-303.