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 $p_1, \dots, p_s \in [x-x^\theta, x + x^\theta]$ are primes, $x = (n/s)^{1/k}$ and $\theta = 0.525 + \epsilon$. We also improve known results for $\theta$ 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 $\theta = 0.55 + \epsilon$. All previously known results on the problem had $\theta > 3/4$.

中文翻译：

---

关于 WARING-GOLDBACH 问题的求和数几乎相等

我们使用转移原理表明，只要 $s$ 取决于 $k \geq 2$ 适当大，每个足够大的自然数 $n$ 满足某些同余条件可以写成 $n = p_1^k + \dots + p_s^k$，其中 $p_1, \dots, p_s \in [xx^\theta, x + x^\theta]$ 是素数，$x = (n/s)^{1/k}$ 和 $ \theta = 0.525 + \epsilon$。当 $k \geq 2$ 和 $s \geq k^2 + k + 1$ 时，我们还改进了 $\theta$ 的已知结果。例如，当 $k \geq 4$ 和 $s \geq k^2 + k + 1$ 我们有 $\theta = 0.55 + \epsilon$。该问题的所有先前已知结果都有 $\theta > 3/4$。