The generalized Waring problem: A new property of positive integers

The paper deals with the problem of whether a positive integer $n>1$ can be written as the sum of $s$ summands that are $r$th powers of integer $s\ge m$, where $m\ge0$ is a chosen integer (for $m=0$ we have the classical Waring problem). For this problem, we define in a natural way arithmetic functions $G(m,r)$ and $g(m,r)$ that are the analogs of the Hilbert functions $G(r)$ and $g(r)$ for the classical Waring problem. It is proved that every positive integer $n$ exceeding some threshold value can be written as the above sum, simultaneously for all $s$, $1\le s\le n$, with a finite number of exceptions, which are determined explicitly.