An additive problem with prime numbers from a thin set (Q1288889)

In the paper an asymptotic formula is given for the number of solutions of the system of equations \(N_i=\sum^k_{j=1} p^i_j\) \((i=1,2,\dots,n)\). Under the condition that the primes \(p_j\) are ``close'' to square numbers in the sense that \(\| \sqrt{p_j}\| \leq p^{-\lambda}\) where \(\lambda>0\) and \(P=N^{1/n}_n\). The theorem is related to the results of \textit{A. Balog} and \textit{J. Friedlander} [Pac. J. Math. 156, 45-62 (1992; Zbl 0726.11061)] and \textit{S. A. Gritsenko} [Russ. Acad. Sci., Izv., Math. 41, 447-464 (1993; Zbl 0810.11057)].