On the lower estimation of non-averaging sets (Q1823986)

A set S of positive integers is non-averaging if the arithmetic mean of two or more members of S never belongs to S. Straus showed that the cardinality f(n) of the largest non-averaging subset of \(\{\) 1,2,...,n\(\}\) satisfies \(f(n)>\exp (c_ 1 \sqrt{\log n}).\) In the other direction, Erdős and Straus proved that \(f(n)<c_ 2 n^{2/3}.\) It is shown that for all sufficiently large n, \(f(n)>c_ 3 n^{1/4}\) for some \(c_ 3>0\). This improves the exponent of 1/5 obtained by \textit{H. L. Abbott} [Acta Math. Hung. 47, 117-119 (1986; Zbl 0607.10039)].