Sets of integers closed under affine operators—the closure of finite sets
Dean G. Hoffman, David A. Klarner · Pacific Journal of Mathematics · 1978
We continue investigation begun in 1974 of sets of integers closed under operators of the form (x 19 •• ,x r )-> m 1 x ι + + m r x r + c f where m lf , m r are integers with gcd(m lf •• ,m r ) = 1.Our main goal here is to prove the following.THEOREM 12. Let r,m lf * ,m r be positive integers, let T be a set of integers, let c be an integer such that (m, + + m r -l)ί + c is positive for each 1e T. If gcd(m 19 -,ra r ) = 1, and if T is closed under the operator (x l9 >",x r ) (x 19 -' ,x^m x Xχ,Λ l-m r x r +c, then the following two statements are equivalent:(1) T is a finite union of infinite arithmetic progressions.(T = ζm i x 1 + + m r x r + c \ A> for some finite set A, where denotes the "smallest" set containing A, and closed under the operator (x 19 •'•,#,•)-» τn ι x 1 + + m r x r + c.