Let $ n \ge 3$ be a prime number and $ a_{1} < a_{2} < \cdots < a_{n}$ be integers. Prove that $ a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $ \{0, 1, 2, \cdots \}$ into sets $ A_{1},A_{2},\cdots,A_{n}$ such that \[ a_{1} + A_{1} = a_{2} + A_{2} = \cdots = a_{n} + A_{n},\] where $ x + A$ denotes the set $ \{x + a \vert a \in A \}$.