On an inverse problem in additive number theory

Let A be a sequence of positive integers and P(A) be the set of all integers which can be represented as the finite sum of distinct terms of A. By improving a result of Hegyvári, Chen and Fang [2] proved that, for a sequence of integers B={b1<b2<⋯}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${B = \{b_{1} < b_{2} < \cdots \}}$$\end{document} , if b1∈{4,7,8}∪{b:b≥11}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${b_{1} \in \{4, 7, 8\} \cup \{b : b \geq 11\}}$$\end{document} and bn+1≥3bn+5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${b_{n+1} \geq 3b_{n} + 5}$$\end{document} for all n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \geq 1}$$\end{document} , then there exists an infinite sequence A of positive integers for which P(A)=N\B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P(A) = \mathbb{N} \setminus B}$$\end{document} ; on the other hand, if b2 =  3b1 +  4, then such A does not exist. In this paper, for b2 = 3b1 +  5, we determine the critical value for b3 such that there exists an infinite sequence A of positive integers for which P(A)=N\B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P(A) = \mathbb{N} \setminus B}$$\end{document} .