Let $k\geq2$ and $s$ be positive integers. Let $\theta\in(0,1)$ be a real
number. In this paper, we establish that if $s>k(k+1)$ and $\theta>0.55$, then
every sufficiently large natural number $n$, subjects to certain congruence
conditions, can be written as $n=p_1^k+\cdots+p_s^k,$ where $p_i(1\leq
i\leq s)$ are primes in the interval
$((\frac{n}{s})^{\frac{1}{k}}-n^{\frac{\theta}{k}},(\frac{n}{s})^{\frac{1}{k}}+n^{\frac{\theta}{k}}]$.
The second result of this paper is to show that if $s>\frac{k(k+1)}{2}$ and
$\theta>0.55$, then almost all integers $n$, subject to certain congruence
conditions, have above representation.Comment: 18 page