Minimal Asymptotic Bases for the Natural Numbers

The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A . Let MhA denote the set of elements that have more than one representation as a sum of h elements of A . It is proved that there exists an asymptotic basis A such that MhA (x) = O(x1-1 11 +E) for every e > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h . It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4 . Several open problems concerning minimal bases are also discussed .