## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  443.10036
Autor:  Erdös, Paul; Graham, Ronald L.
Title:  On bases with an exact order. (In English)
Source:  Acta Arith. 37, 201-207 (1980).
Review:  The statement of the results in this paper require the following definitions: Let A be a set of non-negative integers. A is said to be an asymptotic basis of order r (written ord(A) = r) if r is the least integer such that every sufficiently large integer is expressible as the sum of at most r integers from A (allowing repetition). Also, A is said to have exact order s (written ord^*(A) = s) if s is the least integer for which this is possible with exactly s integers from A. There are basis not having exact order. The following results are established:
I. A basis A = {a1,a2,...} has an exact order if and only if

\gcd{ak+1-ak| k = 1,2,...} = 1.    (1)

II. Let g(r) = max{ord^*(A)} subject to (1), and ord(A) = r. Then,

1/4 (1+0(1))r2 \leq g(r) \leq 5/4 (1+0(1))r2.

Reviewer:  K.Thanigasalam