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

TL;DR

This paper investigates the maximum number of rational-value or zero-sum r-term sums in sets of irrational numbers, providing exact results for certain ranges of r and relating the problem to the Erdős-Ginzburg-Ziv theorem.

Contribution

It determines the maximum counts of rational-value sums and zero-sums for specific r and n, extending the Erdős-Ginzburg-Ziv theorem to new variants.

Findings

Maximum number of rational r-term sums for r<4 or r≥n/2 is established.

Maximum number of zero-sum r-term subsequences in integer sequences is characterized.

Results connect sum problems with classical zero-sum sequence theorems.

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.