The Additive Completion of kth Powers Original Research Article

Let kgreater-or-equal, slanted2 be a fixed integer. For positive integers Mless-than-or-equals, slantN, let Sk(M, N) denote the set of all sets Asubset of[0, M] such that, for all positive integers nless-than-or-equals, slantN, n can be written as n=a+bk with aset membership, variantA and b a positive integer. Define fk(M, N)=min{A: Aset membership, variantSk(M, N)}. Given var epsilon>0, we prove that there exists a δ>0 such that for all sufficiently large Nfk(δN, N)greater-or-equal, slanted(k−var epsilon) N1−1/k.