On bases with an exact order

Introduction . A set A of nonnegative integers is said to be an (asymptotic) basis of order r if every s ufficiently large integer can be expressed . a s a sum of at most r integers taken from A (where repetition is allowed) and r is the least integer with this property . In this case we write ord (A) = r . A basis A is said to have exact order s if every sufficiently large integer is the sum of exactly s elements taken from A (again, allowing repetition) where s is the least integer with this property . We indicate this by writing ord* (A) = s . It is easy to find examples of bases A which. do not have an exact order, e .g., the set of positive odd integers . Of course, if 0 e A and ord (A) = r then ord *(A) = r as well. However, it is not difficult to construct examples of bases A for which