Differences of Signed Representatives: Sharp Answers to Two Questions of Bernert and Arala Santos

Let A ⊂ {−n,…,n}∖{0} contain exactly one of k and −k for every 1 ≤ k ≤ n. At the 2025 Oberwolfach workshop on analytic number theory, C. Bernert and N. Arala Santos asked two questions about such sets. First, must A−A contain (1−o(1))n elements of {1,…,n}? Second, for every B ⊆ {1,…,n} with |B| ≥ εn, must the number of pairs (a,b) ∈ A² with a−b ∈ B be ≫_ε n²? We answer both questions affirmatively, with sharp bounds. 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 sets of the form {1,…,a} ∪ {−(a+1),…,−n}, the second for all |B| ≤ ⌊2n/3⌋+1. The proof is a short double-counting argument. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-14299577-018. Paper page: https://eulersolve.org/papers/owr-14299577-018/