An extremal problem in combinatorial number theory

Abstract

We consider in this paper an extremal problem in combinatorial number theory. Our work is related to the question of difference bases. For given positive integer n, a set of integers 0=a 1 < a 2 <... < a k = n is said to be a difference set for n if every integer v, 0 < v < n, can be written in the form v = a j − a i . Let k(n) denote the minimum value of k over all difference bases for a given integer n. The problem of estimating k(n) was first considered by A. Brauer [1] and subsequently by A. Rényi and L. Rédei [5], P. Erdős and I. S. Gál [3] and J. Leech [4]. The principal result is that k(n) ∼ αn 1/2, as n→ ∞, with a fixed positive constant α. Reasonably good estimates have been obtained for the value of α.