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arXiv · 2603.18805

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

Jing Huang
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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.

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Jing Huang. 2026-03-19. A Proof of a Conjecture of M\'oricz and Nagy on Rational-Value Sums. https://arxiv.org/abs/2603.18805

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