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

DOI: 10.2140/pjm.1978.78.337

Abstract

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

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

1. T is a finite union of infinite arithmetic progressions.
2. T = ⟨m₁x₁ + ⋯ + m_rx_r + c∣A⟩ for some finite set A, where ⟨m₁x₁ + ⋯ + m_rx_r + c∣A⟩ denotes the “smallest” set containing A, and closed under the operator (x₁,⋯,x_r) → m₁x₁ + ⋯ + m_rx_r + c.

Mathematical Subject Classification

Primary: 10L05, 10L05

Milestones

Received: 13 December 1976

Revised: 30 March 1978

Published: 1 October 1978