On a Diophantine inequality over primes. II
Suppose that \(c > 1\) is not an integer. Let \(H(c)\) denote the least integer \(r\) such that for any fixed \(\varepsilon > 0\) and for any \(N > N_0(c, \varepsilon) > 0\), the inequality \[ \left| p_1^c + p_2^c + \dots + p_r^c - N \right| < \varepsilon \] has solutions in prime numbers \(p_1, p_2, \dots, p_r\). This analogue of the Waring--Goldbach problem for fractional exponents was first considered by \textit{I. I. Piatetski-Shapiro} [Mat. Sb. 30, 105--120 (1952; Zbl 0047.28001)]. He proved that when \(c \to \infty\), \(H(c) \leq (4 + o(1))c\log c\) and that \(H(c) \leq 5\) in the range \(1 < c < 3/2\). While the former bound is quite sharp and has withstood improvement to this day, the latter has generated a great deal of recent work, including the paper under review. In this paper, the authors prove (a stronger version of) the inequality \(H(c) \leq 5\) for \(1 < c < 81/40\), \(c \neq 2\). What distinguishes this result from earlier improvements on Piatetski-Shapiro's bound is that the range of \(c\) goes beyond \(c = 2\). The main source for this significant improvement is the use of a result of \textit{O. Robert} and \textit{P. Sargos} [J. Reine Angew. Math. 591, 1--20 (2006; Zbl 1165.11067)] on the number of solutions of certain Diophantine inequality.
scientific article; zbMATH DE number 5145044