Vol. 83, No. 1, 1979

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Sets of integers closed under affine operators—the finite basis theorem

Dean G. Hoffman and David Anthony Klarner

Vol. 83 (1979), No. 1, 135–144
Abstract

This paper is a continuation of investigations of sets T of integers closed under operations f of the form f(x1,,xr) = m1x1 + + mrxr + c, where r, m1,,mr, c are integers satisfying r 2, 0{m1,,mr}, and gcd(m1,,mr) = 1. We have two goals here:

  1. to prove that T = fAfor some finite set A, where fAdenotes the “smallest” set containing A and closed under f, 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.

Mathematical Subject Classification
Primary: 10L05, 10L05
Secondary: 10A35
Milestones
Received: 12 October 1977
Revised: 1 November 1978
Published: 1 July 1979
Authors
Dean G. Hoffman
David Anthony Klarner