**Could one describe the subsets of the integers closed under the binary operation Ax+By 
where A and B are arbitrary fixed integers ?** That is, describe the subsets S 
of the integers such that if $x,y\in S$ then $Ax+By\in S$. Or just the minimal such subsets 
containing 1. 


Do I guess correctly that this question belongs to additive combinatorics ? 



describe subsets of the integers closed under  the binary operation Ax+By 
I think the problem is pretty much solved in a series of papers by Klarner et al; 


David A Klarner and Karel Post, Some fascinating integer sequences. 
A collection of contributions in honour of Jack van Lint. 
Discrete Math. 106/107 (1992), 303–309, MR 93i:11031


D G Hoffman and D A Klarner, Sets of integers closed under affine operators—the finite basis theorem. 
Pacific J. Math. 83 (1979), no. 1, 135–144, MR 83e:10080


D G Hoffman and D A Klarner, Sets of integers closed under affine operators—the closure of finite sets. 
Pacific J. Math. 78 (1978), no. 2, 337–344, MR 80i:10075


