On the Structure of Finite Sets with Identical Representation Functions

• Published in: Frontiers of Mathematics 2024-07, Vol.19 (4), p.665-689
• Main Author: Sun, Cuifang
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics; Research Article
• ISSN: 2731-8648; 2731-8656
• DOI: 10.1007/s11464-022-0220-1

For any positive integer m , let ℤ m be the set of residue classes modulo m . For A ⊆ ℤ m and n ¯ ∈ Z m , let the representation function R A ( n ¯ ) denote the number of solutions of the equation n ¯ = a ¯ + a ′ ¯ with unordered pairs ( a ¯ , a ′ ¯ ) ∈ A × A . For any integer a with ( a, m ) = 1, let ord m ( a ) be the least positive integer h such that a h ≡ 1 (mod m ). Let m = 2 α M , where α is an integer with α ≥ 2 and M is an odd integer with M ≥ 3. In this paper, we prove that if 2 ∣ ord p (2) for some odd prime p with p ≥ M , then there exist two distinct sets A, B ⊆ ℤ m with A ∪ B = ℤ m , ∣ A ∩ B ∣ = 2 and B ≠ A + 2 α − 1 M ¯ such that R A ( n ¯ ) = R B ( n ¯ ) for all n ¯ = Z m . We also prove that if 2 ∤ ord p (2) for any odd prime p with p ∣ M and A, B ⊆ ℤ m with A ∪ B = ℤ m , ∣ A ∩ B ∣ =2, then R A ( n ¯ ) = R B ( n ¯ ) for all n ¯ ∈ Z m if and only if B = A + 2 α − 1 M ¯ .