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Waring's problem for finite intervals


Author: Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 96 (1986), 15-17
MSC: Primary 11P05
DOI: https://doi.org/10.1090/S0002-9939-1986-0813800-3
MathSciNet review: 813800
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Abstract: Let $ f(n,k,s)$ denote the cardinality of the smallest set $ A$ of nonnegative $ k$-th powers such that every integer in $ [0,n]$ is a sum of $ s$ elements of $ A$, and let $ \beta (k,s) = {\text{lim su}}{{\text{p}}_{n \to \infty }}\log f(n,k,s)/\log n$. Clearly, $ \beta (k,s) \geqslant 1/s$. In this paper it is proved that $ f(n,k,s){\text{ < }}c{n^{1/(s - g(k) + k)}}$ for all $ n \geqslant {n_1}(k,s)$, where $ g(k)$ is defined as in Waring's problem, and $ \beta (k,s) \sim 1/s$ as $ s \to \infty $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0813800-3
Keywords: Waring's problem, additive bases, sums of $ k$th powers
Article copyright: © Copyright 1986 American Mathematical Society

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