Sumsets containing a term of a sequence (Q6564327)

For a set \(A\) of integers and a positive integer \(h\ge 2\), define \[ hA=\{a_1+\cdots+a_h:a_1,\cdots,a_h\in A\}. \] A set \(A\) of nonnegative integers is called normal if \(0\in A\) and the greatest common divisor of all elements of \(A\) is 1. In 2006, \textit{H. Pan} [J. Number Theory 117, No. 1, 216--221 (2006; Zbl 1101.11045)] proved the following result, which generalised the results of Lev and Abe:\N\N\textbf{Theorem A}. Let \(k,m,n\ge 2\) be integers. Let \(A\) be a normal subset of \([0,n]\) satisfying \[ |A|>\frac{1}{l+1}((2-\frac{k}{lm})n+2l),\] where \(l=\lceil\frac{k}{m}\rceil\). If \(m\ge 3\), or \(m=2\) and \(k\) is even, then \(kA\) contains a power of \(m\).\N\NIn this paper, the authors extended \textit{H. Pan}'s result [J. Number Theory 117, No. 1, 216--221 (2006; Zbl 1101.11045)] for \(kA(k\ge 3)\) to general sequences and proved that:\N\N\textbf{Theorem 1}. Let \(\beta>2\) be a real number and let \(S=\{s_1,s_2,\cdots\}\) be an unbounded sequence of positive integers such that \(\lim _{n\rightarrow \infty}s_{n+1}/s_n=\alpha<\beta\). Let \(k\ge 3\) be a positive integer. For large enough \(l\), let \(A\) be a normal subset of \([0,l]\) with \(l\in A\) such that \[ |A|\ge\frac{1}{\lambda+1}(2-\frac{k}{\lambda \beta})l,\] where \(l=\lceil\frac{k}{\beta}\rceil\). If \(2<\beta\le 3\), then \(kA\cap S\neq \emptyset\) for all \(k\ge 2\beta/(\beta-2)\). If \(\beta>3\), then \(kA\cap S\neq \emptyset\) for all \(k\ge 3\).\N\N\textbf{Theorem 2}. Let \(\beta>2\) be a real number and let \(S=\{s_1,s_2,\cdots\}\) be an unbounded sequence of positive integers such that \(s_{n+1}/s_n\le\beta\). Let \(k\ge 3\) and \(l\) be positive integers such that \[ l(\frac{k}{\beta}-\lambda+1)\ge \frac{2}{\beta}\lfloor\frac{s_1-1}{2}\rfloor+1.\] let \(A\) be a normal subset of \([0,l]\) with \(l\in A\) satisfying \[ |A|>\frac{2}{\lambda\beta(\lambda+1)} (\lfloor\frac{s_1-1}{2}\rfloor+\frac{\beta}{2}) +\frac{1}{\lambda+1}((2-\frac{k}{\lambda \beta})l+2\lambda),\] where \(\lambda=\lceil\frac{k}{\beta}\rceil\). If \(2<\beta<3\), then \(kA\cap S\neq \emptyset\) for all \(k\ge 2\beta/(\beta-2)\). If \(\beta\ge3\), then \(kA\cap S\neq \emptyset\) for all \(k\ge 3\).