A density version of Waring’s problem
Volume 199 / 2021
Acta Arithmetica 199 (2021), 383-412
MSC: Primary 11P05; Secondary 11B13.
DOI: 10.4064/aa200601-1-2
Published online: 31 May 2021
Abstract
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 a certain constant depending on $k$, with $\mathcal {Z}_k \gt 1$ for every $k$ and $\lim _{k \rightarrow \infty } \mathcal {Z}_k = 1$.
