Waring-Goldbach problem in short intervals (Q6594741)

The author proves new results concerning the Waring-Goldbach problem with almost equal summands. In particular she shows that for \(k \geq 2\) and \(s > k(k+1)\), all sufficiently large integers \(n\) in a suitable residue class can be represented as \N\[\Nn = p_1^k + \dotsb + p_s^k \N\]\Nwith \(p_j \in [(n/s)^{1/k}-n^{\theta/k}, (n/s)^{1/k}+n^{\theta/k}]\), with \(\theta = 0.55\). Furthermore, when either \(k \geq 8\) and \(s > k(k+1)\) or \(2 \leq k \leq 7\) and \(s > 43\) it is shown that one can take \(\theta = 0.525\). The paper contains also results concerning representing almost all \(n\).\N\NThe results improve on previous similar results due to [\textit{J. Salmensuu}, Mathematika 66, No. 2, 255--296 (2020; Zbl 1471.11253)]. Similarly to Salmensuu, the author attacks the problem using a variant of the transference principle. The improvement stems from establishing the pseudorandomness condition on the minor arcs more efficiently. In this part the author utilizes a type I exponential sum estimate of the reviewer and \textit{X. Shao} [Int. Math. Res. Not. 2021, No. 16, 12330--12355 (2021; Zbl 1494.11070)].