Waring–Goldbach problem in short intervals

Let k ≥ 2 and s be positive integers. Let θ ∈ (0, 1) be a real number. In this paper, we establish that if s > k(k + 1) and θ > 0.55, then every sufficiently large natural number n, subject to certain congruence conditions, can be written as

$$n = p_1^k + \cdots + p_s^k,$$

, where p_i (1 ≤ i ≤ s) are primes in the interval \(({({n \over s})^{{1 \over k}}} - {n^{{\theta \over k}}},{({n \over s})^{{1 \over k}}} + {n^{{\theta \over k}}}]\). The second result of this paper is to show that if \(s > {{k(k + 1)} \over 2}\) and θ > 0.55, then almost all integers n, subject to certain congruence conditions, have the above representation.