On Infinite-Difference Sets

Abstract

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.