Sets of integers closed under affine operators—the finite basis theorem
Dean G. Hoffman, David A. Klarner · Pacific Journal of Mathematics · 1979
This paper is a continuation of investigations of sets T of integers closed under operations / of the form f(x lf , x r )= miXx + + m r x r -f c, where r, m lf , m r , c are integers satisfying r Ξ> 2, 0 0 {m lf , m r }, and gcd(m lf , m r ) = 1.We have two goals here:(1) to prove that T= for some finite set A, where denotes the "smallest" set containing A and closed under /, and(2) to show that unless | T | = 1, T is a finite union of infinite arithmetic progressions, either all bounded below, or all bounded above, or all doubly infinite.