We use transference principle to show that whenever $s$ is suitably large depending on $k \geq 2$, every sufficiently large natural number $n$ satisfying some congruence conditions can be written in the form $n = p_1^k + \dots + p_s^k$, where $p_1, \dots, p_s \in [x-x^θ, x + x^θ]$ are primes, $x = (n/s)^{1/k}$ and $θ= 0.525 + ε$. We also improve known results for $θ$ when $k \geq 2$ and $s \geq k^2 + k + 1$. For example when $k \geq 4$ and $s \geq k^2 + k + 1$ we have $θ= 0.55 + ε$. All previously known results on the problem had $θ> 3/4$.

38 pages