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On Infinite-Difference Sets

Published online by Cambridge University Press:  20 November 2018

C. L. Stewart  and
R. Tijdeman
C. L. Stewart
Affiliation:
University of Waterloo, Waterloo, Ontario
R. Tijdeman
Affiliation:
Mathematical Institute, Wassenaarseweg 80, Leiden, Netherlands
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1. Introduction. Let A be a sequence; throughout this paper sequences are understood to be infinite, strictly increasing and composed of non-negative integers. We define D, the infinite-difference set of A, to be the set of those non-negative integers which occur infinitely often as the difference of two terms of A. Plainly D has no positive terms if and only if ai+1 — ai → ∞ as i → ∞. Note that D contains zero. We shall be interested in the case when . Then D certainly contains more than one term. In fact, see Corollary 1, §2, in this case. Here and ḏ denote the (natural asymptotic) upper and lower density respectively.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 5 , 01 October 1979 , pp. 897 - 910
Copyright
Copyright © Canadian Mathematical Society 1979

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