A New Upper Bound for Finite Additive Bases

Let N0 and Z denote the nonnegative integers and integers, respectively, and let |A| denote the cardinality of the set A. Let A be a set of integers, and consider the sumset 2A = {a + a : a, a ∈ A}. Let S be a set of integers. The set A is a basis of order 2 for S if S ⊆ 2A. The set A is called a basis of order 2 for n if the sumset 2A contains the first n nonnegative integers, that is, if A is a basis of order 2 for the interval of integers [0, n− 1] := {0, 1, 2, . . . , n− 1}. We define n(2, A) as the largest integer n such that A is a basis of order 2 for n, that is, n(2, A) = max{n : [0, n − 1] ⊆ 2A}.