On an additive-number-theoretic problem of P. Erdős
For a set \(A= \{a_1,a_2,\dots\}\subseteq\mathbb N\) let NEWLINE\[NEWLINEa_i+ a_j\neq 2^i\quad\text{for all }i,j\in\mathbb N.\tag{\(*\)}NEWLINE\]NEWLINE It is proved:NEWLINENEWLINE (i) For all such sets is \(\underline d(A):= \liminf_{x\to\infty}\, {1\over x} \sum_{a_i\leq x} 1\leq{1\over 2}\).NEWLINENEWLINE (ii) For the set \(\widetilde A= \bigcup^\infty_{j=0} \{3\cdot 2^i\,(\text{mod\,}2^{i+2})\}= \{3,7,11,\dots\}\cup \{6, 14, 22,\dots\}\cup \{12,28, 44,\dots\}\cup\dots\) holds condition \((*)\) and it is \(\underline d(\widetilde A)={1\over 2}\).NEWLINENEWLINE This article arose from a question of P. Erdős.
scientific article; zbMATH DE number 6077890