The Additive Completion of Kth-Powers

Let k ≥ 2 be an integer. For fixed N, we consider a set A N of non-negative integers such that for all integer n ≤ N, n can be written as n = a + b k , a ∈ A N , b a positive integer. We are interested in a lower bound for the number of elements of A N . Improving a result of R. Balasubramanian ( J. Number Theory 29, 1988, 10-12), we prove the following theorem: [formula].