On subset sums of \(r\)-sets (Q685699)

Let \(A\) be a set of distinct integers. \(A\) is called an \(r\)-set if \(A\) has at least \(r\) elements each of which is not divisible by \(q\) for every integer \(q\geq 2\). Given positive integers \(r\) and \(n\), let \(f(n,r)\) denote the maximum cardinality of an \(r\)-set \(A\) such that, for every subset \(B\) of \(A\), \(\sum_{a\in B} a\) is not a power of 2. This is a generalized version of a problem of \textit{P. Erdős} and \textit{G. Freiman} [J. Number Theory 34, 1-12 (1990; Zbl 0697.10047)], who proved that \(f(n)= [n/3]\) for \(n\) large, where \(f(n)\) is the cardinality of a largest set \(A\subseteq \{1,2,\dots, n\}\) such that no power of 2 is the sum of elements of \(A\). In this paper, the author proves that \[ \lim_{r\to\infty} \varlimsup_{n\to\infty} {{f(n,r)} \over n} =0. \]