Vol. 78, No. 2, 1978

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Sets of integers closed under affine operators—the closure of finite sets

Dean G. Hoffman and David Anthony Klarner

Vol. 78 (1978), No. 2, 337–344
Abstract

We continue investigation begun in 1974 of sets of integers closed under operators of the form (x1,,xr) m1x1 + + mrxr + c, where m1,,mr are integers with gcd(m1,,mr) = 1. Our main goal here is to prove the following.

Theorem 12. Let r,m1,,mr be positive integers, let T be a set of integers, let c be an integer such that (m1 + + mr 1)t + c is positive for each t T. If gcd(m1,,mr) = 1, and if T is closed under the operator (x1,,xr)(x1,,xr)m1x1 + + mrxr + c, then the following two statements are equivalent:

  1. T is a finite union of infinite arithmetic progressions.
  2. T = m1x1 + + mrxr + cAfor some finite set A, where m1x1 + + mrxr + cAdenotes the “smallest” set containing A, and closed under the operator (x1,,xr) m1x1 + + mrxr + c.

Mathematical Subject Classification
Primary: 10L05, 10L05
Milestones
Received: 13 December 1976
Revised: 30 March 1978
Published: 1 October 1978
Authors
Dean G. Hoffman
David Anthony Klarner