A hybrid of theorems of Vinogradov and Piatetski-Shapiro
Vinogradov proved that every sufficiently large odd integer is the sum of three primes. A number of Chinese mathematicians [most recently \textit{C. Jia}, Acta Math. Sin. 32, 464-473 (1989; Zbl 0688.10044) and \textit{Zhan}, ibid.] have proved different quantitative versions of the theorem that all large odd integers N can be written as the sum of three ``nearly equal'' primes. In this case the set of primes being used changes with N. \textit{E. Wirsing} [Analysis 6, 285-308 (1986; Zbl 0586.10032)], motivated by work of Erdős and Nathanson, showed the existence of a single thin set S of primes having cardinality \(\sum_{p\in S,p<x}1\ll (x \log x)^{1/3}\), essentially best possible, and such that every large odd integer is the sum of three members of S. Wirsing's proof was based on probabilistic considerations and motivated Wolke at an Oberwolfach meeting (1986) to propose the problem of finding natural explicit thin sets which serve (and exhibiting one such construction). The authors show that such sets are provided by the so-called Piatetski- Shapiro primes. Recall that \textit{I. I. Piatetski-Shapiro} [Mat. Sb., Nov. Ser. 33(75), 559-566 (1953; Zbl 0053.027)] proved that for \(1>\gamma >11/12\) there are infinitely many primes p which are the integer part \(p=[n^{1/\gamma}]\). The sharpest published improvement of this due to \textit{G. Kolesnik} [Pac. J. Math. 118, 437-447 (1985; Zbl 0571.10037)] replaces 11/12 by 34/39. A typical example of the authors' results is the following corollary: Let \(20/21<\gamma_ 1,\gamma_ 2,\gamma_ 3\leq 1\). Then for every large odd N there exist integers \(n_ 1,n_ 2,n_ 3\) such that \([n_ i^{1/\gamma_ i}]=p_ i\) is prime for \(i=1,2,3\) and \(N=P_ 1+p_ 2+p_ 3.\) The proof is achieved by using exponential sum techniques similar to those from the Piatetski-Shapiro problem to reduce the question to the proof of the theorem of Vinogradov itself (which then rests on the Hardy- Littlewood method).
scientific article; zbMATH DE number 4198120