Minimal Bases and g-Adic Representations of Integers

Abstract

Let A be a set of integers, h ≥ 2 an integer. Let hA denote the set of all sums of h elements of A. If hA contains all sufficiently large integers, then A is called an asymptotic basis of order h. An asymptotic basis A of order h is said to be minimal if it contains no proper subset which is again an asymptotic basis of order h. This concept of minimality of bases was first introduced by Stohr [5]. Hartter [1] showed the existence of minimal asymptotic bases by a nonconstructive argument. Nathanson [3] constructed the first nontrivial example of minimal asymptotic bases of order h ≥ 2. Jia and Nathanson [2] recently discovered a simple construction of minimal asymptotic bases of order h ≥ 2 by using powers of 2. Furthermore, for any α: 1/h ≤; α < 1, they constructed a minimal asymptotic basis A of order h such that x α < A(x) < x α, where A(x) is the number of positive elements not exceeding x. In the present paper, we shall generalize these results to g-adic representations of integers.