For any positive integer m, let $\mathbb {Z}_{m}$ be the set of residue classes modulo m. For $S\subseteq \mathbb {Z}_{m}$ and $\overline{n}\in \mathbb {Z}_{m}$, let the representation function $R_{S}(\overline{n})$ denote the number of solutions of the equation $\overline{n}=\overline{s}+\overline{s'}$ with unordered pairs $(\overline{s}, \overline{s'})\in S \times S$ and $\overline{s}\ne \overline{s'}$. For any integer a with $(a, m)=1$, let $\mathrm{{ord}}_{m}(a)$ be the least positive integer h such that $a^{h}\equiv 1\pmod {m}$. Let $m=2^{\alpha }M \ge 4$, where $\alpha $ is a positive integer and M is a positive odd integer. In this paper, we prove that if $M=1$ and $\alpha =2$ or $M\ge 3$ and $2\mid \mathrm{{ord}}_{p}(2)$ for some odd prime p with $p\mid M$, then there exist two distinct sets $A, B\subseteq \mathbb {Z}_{m}$ with $|A\cup B|=m-2, |A\cap B|=0$ and $B\ne A+\overline{\frac{m}{2}}$ such that $R_{A}(\overline{n})=R_{B}(\overline{n})$ for all $\overline{n}\in \mathbb {Z}_{m}$. We also prove that if $M=1$ and $\alpha \ge 3$ or $M\ge 3$ and $2\not \mid \mathrm{{ord}}_{p}(2)$ for any odd prime p with $p\mid M$ and $A, B\subseteq \mathbb {Z}_{m}$ with $|A\cup B|=m-2$, $|A\cap B|=0$, then $R_{A}(\overline{n})=R_{B}(\overline{n})$ for all $\overline{n}\in \mathbb {Z}_{m}$ if and only if $B=A+\overline{\frac{m}{2}}$.
| [1] |
Chen SQ, Chen YG. Integer sets with identical representation functions II. Eur. J. Combin., 2021, 94. 103293
|
| [2] |
Chen SQ, Wang RJ, Yu WX. On the structure of sets in a residue class ring with the same representation function. Adv. Appl. Math., 2023, 148. 102533
|
| [3] |
Chen SQ, Yan XH. On certain properties of partitions of $\mathbb{Z} _m$ with the same representation function. Discrete Math., 2020, 343. 111981
|
| [4] |
Chen YG, Lev VF. Integer sets with identical representation functions. Integers, 2016, 16: A36
|
| [5] |
Dombi G. Additive properties of certain sets. Acta Arith, 2002, 103: 137-146.
|
| [6] |
Duan ZX, Sun CF. On additive properties of two sets in a residue class ring. Ramanujan J., 2025, 66: 67.
|
| [7] |
Kiss SZ, Sándor C. Partitions of the set of nonnegative integers with the same representation functions. Discrete Math., 2017, 340: 1154-1161.
|
| [8] |
Lev VF. Reconstructing integer sets from their representation functions. Electron. J. Combin., 2004, 11: R78.
|
| [9] |
Sándor C. Partitions of natural numbers and their representation functions. Integers, 2004, 4: A18
|
| [10] |
Sun CF. On the structure of finite sets with identical representation functions. Front. Math., 2024, 19: 665-689.
|
| [11] |
Sun CF, Cheng Z. On the structure of finite sets with the same representation functions. J. Number Theory, 2022, 241: 352-363.
|
| [12] |
Sun CF, Cheng Z. Partitions of $\mathbb{Z} _m$ with identical representation functions. Discrete Appl. Math., 2025, 362: 1-10.
|
| [13] |
Sun CF, Pan H. On the structure of finite sets with identical representation functions. Bull. Iran. Math. Soc., 2023, 49: 35.
|
| [14] |
Tang M. Partitions of the set of natural numbers and their representation functions. Discrete Math., 2008, 308: 2614-2616.
|
| [15] |
Tang M, Li JW. On the structure of some sets which have the same representation functions. Period. Math. Hungar., 2018, 77: 232-236.
|
| [16] |
Yang QH, Chen FJ. Partitions of $\mathbb{Z} _{m}$ with the same representation functions. Australas. J. Combin., 2012, 53: 257-262
|
| [17] |
Yang QH, Tang M. Representation functions on finite sets with extreme symmetric differences. J. Number Theory, 2017, 180: 73-85.
|
Funding
National Natural Science Foundation of China(12371003)
Rights & permissions
School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature
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