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| Representation Functions on the Additive Group of Residue Classes |
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Citation： |
Cuifang SUN.Representation Functions on the Additive Group of Residue Classes[J].Chinese Annals of Mathematics B,2025,46(2):233~240 |
| Page view： 450
Net amount： 177 |
Authors： |
Cuifang SUN; |
Foundation： |
the National Natural Science Foundation of China (No. 12371003) |
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| Abstract： |
For any positive integer m, let Zm be the additive group of residue classes
modulo m. For A ? Zm and n ∈ Zm, let the representation function RA(n) denote the
number of solutions of the equation n = a + a′ with unordered pairs (a, a′) ∈ A × A. Let
m = 2αM > 2, where α is a positive integer and M is a positive odd integer. In this
paper, the author proves that if M ≥ 3, then there exist two distinct sets A, B ? Zm with
|A∪B| = m? 2, A∩B = ? and B 6= m2 + A such that RA(n) = RB(n) for all n ∈ Zm. The
author also proves that if M = 1 and A, B ? Zm with |A ∪ B| = m ? 2 and A ∩ B = ?,
then RA(n) = RB(n) for all n ∈ Zm if and only if B = m2 + A. |
Keywords： |
S´arkÖzy’s problem Representation function Residue class |
Classification： |
11B34 |
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