A remark on infinite arithmetic progressions (Q801111)

Write \(A(a,b)=\{a+\lambda b| \lambda \in {\mathbb{N}}\}\) for the set of terms of the infinite arithmetic progression with first term \(a\in {\mathbb{N}}\) and common difference \(b\in {\mathbb{N}}\setminus \{0\}\). For \(X\subseteq {\mathbb{N}}\), write \(A_ X(a,b)=\{a+\lambda b| \lambda \in X\}\). The following result is proved: There exists an infinite recursive subset X of \({\mathbb{N}}\) such that \(| A_ X(a,b)\cap A_ X(c,d)| \leq 2\) whenever A(a,b)\(\neq A(c,d)\).