The additive completion of kth powers

Let k greater than or equal to 2 be a fixed integer. For positive integers M less than or equal to N, let S-k(M, N) denote the set of all sets A subse t of [0, M] such that, for all positive integers n less than or equal to N, n can be written as n = a + b(k) with a is an element of A and b a positiv e integer. Define f(k)(M, N) = min{\A\: A is an element of S-k(M, N)}. Give n epsilon > 0, we prove that there exists a delta > 0 such that for all suf ficiently large N f(k)(delta N, N) greater than or equal to (k - epsilon) N1-1/k. (C) 1999 Academic Press.