Maximizing the number of rational-value sums or zero-sums

Abstract

What is the maximum number of $r$-term sums admitting rational values in $n$-element sets of irrational numbers? We determine the maximum when $r<4$ or $r\geq n/2$ and also in case when we drop the condition on the number of summands. It turns out that the $r$-term sum problem is equivalent to determine the maximum number of $r$-term zero-sum subsequences in $n$-element sequences of integers, which can be seen as a variant of the famous Erd\H{o}s-Ginzburg-Ziv theorem.