On the Waring-Goldbach problem with almost equal summands

We use transference principle to show that whenever $s$ is suitably large depending on $k\ge 2$, 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$p_1, \dots, p_s \in [x-x^\theta, x + x^\theta]$areprimes,$x = (n/s)^{1/k}$and$\theta = 0.525 + \epsilon$\mathrm{.}Wealsoimproveknownresultsfor$\theta$when$k \geq 2$and$s \geq k^2 + k + 1$\mathrm{.}Forexamplewhen$k \geq 4$and$s \geq k^2 + k + 1$wehave$\theta = 0.55 + \epsilon$. All previously known results on the problem had $\theta >3\mathrm{/}4$.