On exponential type sequences (Q6110606)

Given a multiset \(S\) of integers, let also \(P(S)\) be the set of all finite sums of distinct terms taken from \(S\); in particular \(0 \in P(S)\). Fix an integer \(p\ge 2\) and a nonempty finite multi \(A\) of positive integers. Define the multiset \[ S_pA:=\{p^i a: i \ge 0, a \in A\}. \] In the manuscript, the authors prove the following results: \begin{itemize} \item[1.] If \(P(A)\) contains a subset \(\{c_1,\ldots,c_p\}\) such that \(\{c_1/d, \ldots, c_p/d\}=\{1,\ldots,p\}\bmod{p}\), where \(d:=\mathrm{gcd}(c_1,\ldots,c_p)\), then \(P(S_pA)\) has positive lower asymptotic density. \item[2.] If \(P(S_pA)\) is cofinite, then \(P(A)\) contains a complete a subset \(\{c_1,\ldots,c_p\}\) such that \(\{c_1, \ldots, c_p\}=\{1,\ldots,p\}\bmod{p}\). \item[3.] If \(P(S_pA)\) is cofinite and there exists \(a \in P(A)\) with \(p\nmid a-b\) for all \(b \in P(A)\setminus \{a\}\), then \(P(A)\) contains \(\{0,1,\ldots,p-1\}\). \end{itemize} Lastly, they provide a sufficient condition on \(S_pA\) which ensures that \(P(S_pA)=\mathbb{N}\).