A Density version of Waring's problem

Abstract

In this paper, we study a density version of Waring's problem. We prove that a positive density subset of $k$th-powers forms an asymptotic additive basis of order $O(k^2)$ provided that the relative lower density of the set is greater than $(1-{\mathcal{Z}}_k^{-1}/2{)}^{1/k}$, where ${\mathcal{Z}}_k$ is certain constant depending on $k$ for which it holds that ${\mathcal{Z}}_k>1$ for every $k$ and ${\lim }_{k\to \infty }{\mathcal{Z}}_k=1$.

26 pages; Correction of the proof of Theorem 1.3 (The problem in the proof occurs in the published version)