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

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.

💡 Research Summary

The paper settles a conjecture of Móricz and Nagy concerning the maximal number of r‑element subsets whose sum is rational in an n‑element set of irrational numbers. The authors first translate the “rational‑sum” condition into a zero‑sum condition by means of a Q‑linear map whose kernel is exactly Q. Lemma 2.1 constructs such a map φ:V→ℝ on the Q‑vector space V generated by the given irrational set together with 1, guaranteeing that φ(a_i)≠0 for each element a_i and that a subset has rational sum if and only if the corresponding φ‑images sum to zero. Consequently, the original extremal problem becomes: given non‑zero real numbers x_1,…,x_n, how many r‑subsets can have sum zero?

To bound this number, the authors introduce a partially ordered set X consisting of pairs (S,T) where S⊆P (indices of positive terms) and T⊆N (indices of negative terms) with |S|+|T|=r. The order (S,T)⪯(S′,T′) holds when S⊆S′ and T⊇T′. For any zero‑sum r‑subset I, the pair (P(I),N(I)) belongs to X, and distinct zero‑sum subsets give incomparable elements; thus the collection A of such pairs forms an antichain in X.

Lemma 2.2 exploits the antichain property via a probabilistic chain argument. By taking uniform random permutations of P and N, a random chain C(ω) is built. Since an antichain and a chain intersect in at most one element, the expected size of A∩C(ω) is at most 1. Computing the exact probability that a fixed pair (S,T) lies on the random chain yields the inequality

|Z_r(x)| ≤ max_{1≤t≤r−1} C(q,t)·C(p,r−t),

where p=|P|, q=|N|, and t=|T| is the number of negative terms in the subset. This reduces the problem to a purely combinatorial maximization over p,q.

Lemma 2.3 carries out this maximization. For each fixed t, the function F_t(q)=C(q,t)C(n−q,r−t) is shown to be unimodal in q, attaining its maximum at q_t=⌊t(n+1)/r⌋. Substituting q_t and simplifying, the authors prove that the overall maximum over all admissible q and t equals

m·C(n−m, r−1),

where m=⌊n/r⌋. The proof uses careful algebraic manipulation, Vandermonde’s identity, and bounding of binomial coefficients.

Having established the universal upper bound, the authors verify its sharpness by reproducing the construction originally given by Móricz and Nagy. Choose an irrational α and distinct rationals u_1,…,u_m and v_1,…,v_{n−m}. Define

b_i = u_i − (r−1)α (i=1,…,m), c_j = v_j + α (j=1,…,n−m).

All b_i and c_j are irrational and distinct. An r‑subset of the resulting set has rational sum precisely when it contains exactly one b_i and r−1 of the c_j’s; there are exactly m·C(n−m, r−1) such subsets. Hence the upper bound is attained, confirming the conjecture for all 1<r<n.

Corollary 1.2 translates the result back to sequences of non‑zero rational (and integer) numbers. For any sequence x∈(ℚ{0})^n, the number Z_x(r) of r‑term zero‑sum subsequences satisfies

|Z_x(r)| ≤ m·C(n−m, r−1),

and this bound is tight. An explicit integer sequence achieving equality is

y = (−(r−1),…,−(r−1)︸m times, 1,…,1︸n−m times),

where a zero‑sum r‑subsequence occurs exactly when it contains one −(r−1) and r−1 ones.

In summary, the paper combines a linear‑algebraic reduction, an antichain‑chain probabilistic argument, and a precise binomial maximization to prove that the construction of Móricz and Nagy is optimal in every case. This resolves the conjecture completely and provides the exact extremal value for the maximal number of zero‑sum r‑subsequences in both rational and integer sequences, enriching the theory of extremal zero‑sum problems with a new “maximum‑count” perspective.