On representation problems in the additive number theory
Let \(P(A)=\{\sum \varepsilon_ia_i\mid \varepsilon=0\text{ or }1\); \(a_i\in A\); \(\sum\varepsilon_i<\infty\}\) if \(A=\{a_1<a_2<\dots \}\subseteq\mathbb{N}\). In the first result of the paper the author improves an unpublished result of S. A. Burr proving that if \(B=\{7\leq b_1<b_2<\dots\}\) with \(b_{n+1}\geq 5b_n\) for every \(n\), then there exists a sequence of integers \(A\) with \(P(A)=\mathbb{N}\setminus B\). The second result proves that this cannot be improved essentially, for if \(B\) is a Sidon sequence with \(b_{n+1}/b_n<\sqrt{2}\) then there is no such sequence \(A\). The final result answers a representation question posed by \textit{P. Erdös} and \textit{E. Szemerédi} [Studies in Pure Mathematics, Mem. of P. Turán, 213-218 (1983; Zbl 0526.10011)], namely: If \(n\) is a positive integer and \(t\) an integer satisfying \(\binom{n+1}{2}+1\leq t\leq 2^n\), then there exists an \(A=\{a_1<a_2<\dots a_n\}\) for which \(\left |P(A)\right |=t\).
scientific article; zbMATH DE number 923732