# Verification report — OWR-14299577-018 (Bernert and Arala Santos, Oberwolfach Report 51/2025)

Verification date: 2026-09-28.

**Verdict.** Complete affirmative answer to both questions, with sharp bounds.
- Let A ⊂ {−n,…,n}∖{0} contain exactly one of k and −k for every 1 ≤ k ≤ n. Then at most two elements of [1,n] are
  missing from A − A.
- For every B ⊆ [1,n], the number of pairs (a,b) ∈ A² with a − b ∈ B is at least ⌊(|B|−1)²/4⌋.
- Both bounds are attained by the sets A_a = {1,…,a} ∪ {−(a+1),…,−n}, the second for all |B| ≤ ⌊2n/3⌋+1.

Unrefereed.

## Statement checked
- **Source.** Oberwolfach Reports 22 (2025), no. 4 (Report 51/2025, *Analytic Number Theory*), summary of the
  problem session compiled by T. F. Bloom, p. 2756, DOI 10.4171/OWR/2025/51. The problem was proposed by Christian
  Bernert and Nuno Arala Santos. The page was rendered and read.
- **Corpus record.** ulamai/UnsolvedMath, OWR-14299577-018, status `open`, `proposed_by` empty.
  - The `statement` field matches the source.
  - The `original_statement` field is garbled: it drops the set-up sentence and splices in two sentences from the
    preceding problem.
- **Reading.** The natural reading was adopted: A has size n and contains exactly one of ±k for each k. The first
  referee checked the alternatives and found none under which the first question becomes harder:
  - counting with multiplicity;
  - the k / 1−k variant, which the source answers "no";
  - dropping |A| = n;
  - misprint readings.

## Proof (in the paper)
- **Notation.** Write g for the ±1 indicator of A, which is odd, and D(d) for the set of positions x with
  g(x) = g(x+d).
- **Lemma 2.1.** |D(d)| = 2r(d), and the mirror x ↦ −x−d is a fixed-point-free involution of D(d).
- **Lemma 2.2.** For d < d′ of the same parity and j = (d′−d)/2, exactly one of j ∈ D(d) and −j ∈ D(d′) holds.
- **Proposition 1.3.** Charging each pair to one of these defects gives Σ_{d∈S} r(d) ≥ C(|S|,2) for every set S of
  one parity.
- **Theorems.** Theorem 1.1 is the case |S| = 2. Theorem 1.2 follows by splitting B into its odd and even parts.
- **Proposition 3.1** computes r for the two-block sets A_a and gives both equality cases.

## Computations (exact; scripts and outputs in reproducibility/)
- **Finder.** Theorem 1.1 for n ≤ 20.
- **First referee.** Theorem 1.1 for n ≤ 26. The extremal pairs were computed exhaustively for n ≤ 26 and with a
  separate solver for n ≤ 64. The minima of the sum of the m smallest r(d) equal ⌊(m−1)²/4⌋ exactly when
  m ≤ ⌊2n/3⌋+1, checked for n ≤ 26. Lemma checks on 69 test sets up to n = 120.
- **Second referee (final paper).** 36 checks, all passing:
  - every clause of the lemmas and of the charging;
  - Proposition 1.3 and Theorem 1.2 exhaustively for n ≤ 26;
  - Proposition 3.1 and the equality cases for n ≤ 40.
- **Author.** Proposition 1.3 and Theorems 1.1 and 1.2 for all signed sets with n ≤ 20.

## Independent audits
| Round | Item | Verdict |
|---|---|---|
| 1 (first version) | Statement fidelity | CONFIRMED_WITH_FIXES (attribution corrected to Bernert and Arala Santos) |
| 1 | Proof of Question 1 | CONFIRMED |
| 1 | Answer as posed | CONFIRMED_WITH_FIXES (Question 2 is also answered yes; the referee's own double count gave (|B|−1)(|B|−2)/8) |
| 1 | Novelty | CONFIRMED_WITH_FIXES (no earlier treatment found) |
| 2 (final paper) | Lemmas, Proposition 1.3, Theorems 1.1–1.2, equality cases | CONFIRMED |
| 2 | Proposition 3.1; Remark 3.2(1); verification section | CONFIRMED_WITH_FIXES (all fixes applied) |

The lead auditor sharpened the double count to Proposition 1.3. The second referee confirmed it and extended the
proved equality range to ⌊2n/3⌋+1. The computations show that this range cannot be improved for n ≤ 26.

## Relation to the literature, novelty and scope
- **Citations.** The report (published 2026-03-04) has no citing works in Crossref or OpenAlex.
- **Search.** No treatment of either question was found on arXiv, in later papers of the proposers, or on
  erdosproblems.com. The closest classical relative is Motzkin's problem on sets with missing differences (Cantor–Gordon,
  1973).
- **Caveat.** The arguments are elementary, and the answers may be known informally. This negative search is not a
  proof of priority.
- **Open.** Remark 3.2(2), the characterisation of the extremal sets, is computational only.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/