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A Proof of a Conjecture of Móricz and Nagy on Rational-Value Sums

Jing Huang

Abstract

Móricz 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.

A Proof of a Conjecture of Móricz and Nagy on Rational-Value Sums

Abstract

Móricz and Nagy introduced the problem of maximizing the number of -element subsets with rational sums in an -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 -element set of irrational numbers containing exactly such subsets, where . They conjectured that this construction is always optimal for any . 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 -term zero-sum subsequences in sequences of nonzero integers.
Paper Structure (2 sections, 5 theorems, 76 equations)

This paper contains 2 sections, 5 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem']} and Corollary \ref{['cor:problem12']}

Key Result

Theorem 1.1

Let $n>1$ be a positive integer, $1<r<n$, and put $m=\left\lfloor \frac{n}{r}\right\rfloor.$ Then every $n$-element set $A\subseteq \mathbb{R}\setminus\mathbb{Q}$ satisfies We further have

Theorems & Definitions (10)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['theorem']}
  • proof : Proof of Corollary \ref{['cor:problem12']}