On a ternary Diophantine inequality over primes

Let 1<c<21097,c≠2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< c < \frac{210}{97}, c \ne 2$$\end{document}. In this paper, it is proved that for every sufficiently large real number N, for almost all real R∈(N,2N]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\in (N, 2N]$$\end{document}, the Diophantine inequality |p1c+p2c+p3c-R|<log-1N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |p_1^c + p_2^c + p_3^c - R| < \log ^{-1}N \end{aligned}$$\end{document}is solvable in primes 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}. Moreover, we prove that the Diophantine inequality |p1c+p2c+p3c+p4c+p5c+p6c-N|<log-1N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |p_1^c + p_2^c + p_3^c + p_4^c + p_5^c + p_6^c - N| < \log ^{-1}N \end{aligned}$$\end{document}is solvable in primes p1,p2,p3,p4,p5,p6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1,p_2,p_3,p_4,p_5,p_6$$\end{document}. This result constitutes a refinement upon that of Cai (Int J Number Theory 14:2257–2268, 2018).