On a Problem of Partitions of ℤ 𝑚 with Identical Representation Functions
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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.
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References
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