A Proof of a Conjecture of M\'oricz and Nagy on Rational-Value Sums

TL;DR

This paper proves a conjecture by Móricz and Nagy, confirming the optimality of a specific construction for maximizing rational-sum subsets in irrational sets, and determines the maximum number of zero-sum subsequences in integer sequences.

Contribution

It confirms the conjecture that a known construction is always optimal for maximizing rational-sum subsets, and extends results to zero-sum subsequences in integer sequences.

Findings

Confirmed the conjecture for all cases.

Determined the exact maximum number of zero-sum subsequences.

Combined order-theoretic and combinatorial optimization techniques.

Abstract

M\'oricz and Nagy introduced the problem of maximizing the number of $r$-element subsets with rational sums in an $n$-element set of irrational numbers, and showed that it is equivalent to an extremal zero-sum problem. They determined the exact maximum in several cases. For the remaining range, they presented an explicit construction of an $n$-element set of irrational numbers containing exactly $m\binom{n-m}{r-1}$ such subsets, where $m=\lfloor n/r\rfloor$. They conjectured that this construction is always optimal for any $1<r<n$. In this paper, we confirm that conjecture. Our proof combines an order-theoretic antichain argument for zero-sum subsets with a sharp maximization of the resulting binomial expressions. As a consequence, we determine exactly the maximum number of $r$-term zero-sum subsequences in sequences of $n$ nonzero integers.