Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  359.10045
Autor:  Erdös, Paul; Newman, D.J.
Title:  Bases for sets of integers. (In English)
Source:  J. Number Theory 9, 420-425 (1977).
Review:  If every member of a set A of positive integers can be expressed in the form b+b' for some b, b' in B, then the set B of non-negative integers is called a basis for A. The problem is to find bounds on the size m of the smallest basis B for a set A of type (n,N), i.e. for a set A with n elements, the largest of which is N. It is easy to show that n ½ \leq m \leq max{n+1,(4N+1) ½ }. The authors show that, for most sets A of type (n,N),

m > max\left{\frac n{log N},\frac{\sqrt N}2\right},

so that, for most sets A, the actual value of m is closer to the upper bound than to the lower bound. A special study is then made of the sequence of squares, A = {12,22,...,n2}, and it is shown that this sequence is untypical since, for arbitrarily large M, n2/3-\epsilon \leq m \leq n/ log mn. (There is an unfortunate misprint at this point.) Closing remarks include the observation that the value of m depends critically not just on n and N, but on the arithmetical character of the sequence.
Reviewer:  I.Anderson
Classif.:  * 11B13 Additive bases
                   11B83 Special sequences of integers and polynomials

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