The three primes theorem with primes in the intersection of two Piatetski--Shapiro sets

The well-known three primes theorem says that, for every sufficiently large odd integer N, the equation N=p1+p2+p3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=p_1+p_2+p_3$$\end{document} is solvable for prime variables p1,p2,p3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1, p_2, p_3$$\end{document}. In this paper we shall prove that the three primes theorem still holds if each of the three primes is in the intersection of two Piatetski--Shapiro sets.