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In additive number theory, the set A of nonnegative integers is an asymptotic basis of order 2 if every sufficiently large integer can be written as the sum of two elements of A . Let rA (n) denote the number of representations of n in the form n = a+a', where a, a' eA and a < a' . An asymptotic basis A of order 2 is minimal if no proper subset of A is an asymptotic basis of order 2 . Erdös and Nathanson [2] proved that if A is an asymptotic basis of order 2 such that rA (n) > c -log n for some constant c > 1/log (4/3) and every sufficiently large integer n, then some subset of A is a minimal asymptotic basis of order 2. It is an open problem to determine whether the set A must contain a minimal asymptotic basis of order 2 if r, (n) merely tends to infinity as n tends to infinity . This paper contains several results connected with this question . Let ISI denote the cardinality of the set S . For any set A of nonnegative integers, let

Additive bases with many representations