Let $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x\ $ with some conditions$\ \}$.    
Let $B=\mathbb Z_{\ge 0}-A$.    
Define $\ 2A= \{a+b : a \in A,\ b \in A\}$.    
Define $\ 2B=\{a+b : a \in B,\ b \in B\}$.   
Then the set $\ \{n,\ n+1k ,\ n+2k, \ ...\}\ \subseteq\ 2A\ $or $\ 2B$ for some positive  integers $n,k$?