Waring's problem with almost proportional summands

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 $$\mathrm{\mid }{x}_i^n-{\mu }_{i}N\mathrm{\mid }\le H,H\ge N^{1-\theta (n,r)+\epsilon },\theta (n,r)=\frac{2}{(r+1)(n^2-n)},$$ where ${\mu }_1,\dots ,{\mu }_r$ are positive fixed numbers, and ${\mu }_1+\dots +{\mu }_n=1$. This result strengthens the theorem of E.M.Wright.