Waring’s problem for finite intervals

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 $s \to \infty$.