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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 ?

I only know the (easy) answers for A=B (an arithmetical progression) and A=2,B=1 (where it happens to be the union of two arithmetical projections).

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 ?

I only know the (easy) answers for A=B (an arithmetical progression) and A=2,B=1 (where it happens to be the union of two arithmetical projections).

1

describe subsets of the integers closed under the binary operation Ax+By

Could one describe the subsets of the integers closed under the binary operation Ax+By where A and B are arbitrary integers ?

Do I guess correctly that this question belongs to additive combinatorics ? I only know the (easy) answers for A=B (an arithmetical progression) and A=2,B=1 (where it happens to be the union of two arithmetical projections).