Waring's problem with almost proportional summands

Abstract

For $n\ge 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r=2^n+1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i=\overline{1,r}$, satisfying the conditions $$|x_i^n-\hat{I}{¼}_{i}N|\le H,H\ge N^{1-\hat{I}¸(n,r)+\epsilon },\hat{I}¸(n,r)=\frac{2}{(r+1)(n^2-n)},$$ where $\hat{I}{¼}_1,\dots ,\hat{I}{¼}_r$ are positive fixed numbers, and $\hat{I}{¼}_1+\dots +\hat{I}{¼}_n=1$. This result strengthens the theorem of E.M.Wright.

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