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A density version of Waring’s problem

Volume 199 / 2021

Juho Salmensuu 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 $\lim _{k \rightarrow \infty } \mathcal {Z}_k = 1$.

Authors

  • Juho SalmensuuDepartment of Mathematics and Statistics
    University of Turku
    FI-20014 University of Turku, Finland
    e-mail

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