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

Let \(A\subset\{-n,\dots,n\}\setminus\{0\}\) contain exactly one of \(k\) and \(-k\) for every \(1\le k\le n\). 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 \(\{1,\dots,n\}\)? Second, for every \(B\subseteq\{1,\dots,n\}\) with \(|B|\ge\varepsilon n\), must the number of pairs \((a,b)\in A^2\) with \(a-b\in B\) be \(\gg_\varepsilon n^2\)? We answer both questions affirmatively, with sharp bounds. At most two elements of \(\{1,\dots,n\}\) are missing from \(A-A\). For every \(B\subseteq\{1,\dots,n\}\), the number of pairs \((a,b)\in A^2\) with \(a-b\in B\) is at least \(\lfloor(|B|-1)^2/4\rfloor\). Both bounds are attained by sets of the form \(\{1,\dots,a\}\cup\{-(a+1),\dots,-n\}\), the second for all \(|B|\le\lfloor 2n/3\rfloor+1\). The proof is a short double-counting argument. This is an unrefereed note.