Partitions of Residue Classes Modulo m with Identical Representation Functions

Abstract

Background and Objectives: The problem of identical representation functions has long been an interesting question in additive number theory. This study is motivated by Sárközy’s problem, which asks whether there exist two distinct subsets of integers that have the same representation function. Early works by Nathanson (1978) introduced the fundamental concept of the representation function, while later studies by Erdős et al., (1986) formulated the well-known question on additive properties of integer sequences. Dombi (2002) provided partial solutions by proving that for certain sets of integers no such pair of subsets exists, but for some specific cases, such subsets can indeed be found. Subsequent research by Chen and Wang (2003), and later by Yang and Chen (2012), extended the problem to modular arithmetic within the residue class ring modulo  (denoted as . ).

This paper continues and deepens this line of research by focusing on the partition problem of the residue class ring ​ . in which two subsets  A and  B  satisfy for all , ​ The goal is to determine necessary and sufficient conditions under which two subsets of ​ with  ​  have identical representation functions. The main objective is to determine necessary and sufficient conditions ensuring the equality of representation functions between the two subsets, with special attention to the case where  is an odd integer. This investigation aims to clarify the structural and symmetric properties that govern additive equivalence in modular systems.

Methodology: The methodology integrates algebraic reasoning, modular arithmetic, and additive number theory. The representation function   for a subset ​  .  counts the number of unordered pairs       such that  . The analysis proceeds by defining this function explicitly, followed by constructing a characteristic function     to represent membership in    An illustrative example for    is first presented to demonstrate how representation values are computed and to visualize symmetry among residue classes under modular addition. Afterward, the study systematically distinguishes four fundamental cases according to the relative configurations of the subsets    and   within ​ These four cases reflect different structural relationships between and , expressed as:

1.     and
2.   and
3.   and
4.   and

For each case, the study employs case analysis combined with modular congruence arguments to determine whether equality of representation functions is maintained. The methodology also introduces Lemma 2.1, asserting that if  is odd, then    implies   This lemma ensures injectivity in modular doubling and serves as a key tool in verifying equivalence of representations. The overall analytic process combines theoretical proofs, parity-based reasoning, and direct algebraic verification under each of the four cases.By employing additive number theory and properties of modular arithmetic, the study analyzes the representation function     through the use of characteristic functions and modular congruences. A series of lemmas and theorems are developed to characterize relationships between the subsets and Logical deductions and case analyses are used to handle different configurations of subsets, distinguishing between cases where elements satisfy symmetric and congruence relations in . 

Main Results: The results demonstrate that if the representation functions of  and are identical for all  ,    then   must be odd. In this paper, we give a necessary and sufficient condition for two subsets   and of    such that     for all  .   Moreover, the equality of representation functions can be completely characterized through the following four fundamental cases, each describing a distinct modular relationship between    and

Case 1, Equality holds precisely when        and

This configuration represents a pure modular shift, in which one subset can be obtained from the other through a fixed translation modulo .

Case 2, Equality is achieved when         and

Here, the equality of representation functions results from reflection about a modular axis, meaning each element of corresponds to a symmetric counterpart in under modular equivalence.

Case 3, Complementary partitions of ​ preserve representation equality if the subset cardinalities satisfy 
 and 

Here, the subsets  and together form a balanced bipartition of ,​ ensuring exact equality of additive representations across all residue classes.

Case 4, Equality also holds in a mixed configuration combining both shift and reflection properties, where 
  and .

This case demonstrates that the equality of representation functions can arise from a hybrid symmetry, where the subsets are simultaneously related by a modular shift and a reflection, depending on the specific overlap pattern among elements modulo .

Conclusions: This research enhances the understanding of additive partitions in modular systems by establishing precise algebraic criteria for identical representation functions. The results emphasize that modular symmetry, reflection, and parity are decisive factors for representation equality. For odd moduli, subsets and  that are structurally complementary under modular addition always yield identical representation functions. These findings not only extend classical additive number theory but also suggest future directions for study. Possible extensions include examining even moduli where symmetry conditions are broken, exploring multidimensional modular structures such as and applying computational algorithms to test large-scale modular partitions.Ultimately, this study provides a rigorous theoretical foundation for understanding how additive identities and modular symmetries interact within residue class rings, contributing both to pure mathematics and to the broader combinatorial analysis of algebraic structures.