A convex set with a rich difference (Q2681281)

A finite set \(A=\{a_1<a_2<\cdots<a_n\}\subset\mathbb{R}\) is said to be convex if \(a_i-a_{i-1}<a_{i+1}-a_i\) holds for all \(2\le i\le n-1\). The purpose of this paper is to give a construction of a convex set with a rich difference which improves the construction of \textit{T. Schoen} [Can. Math. Bull. 57, No. 4, 877--883 (2014; Zbl 1375.11010)]. It is proved that for every \(m\in\mathbb{N}\), there exists a convex set \(A\subset\mathbb{R}\) of size \(2 m\) and a non-zero element \(d\in A-A:=\{a-b\vert a,b\in A\}\) such that \(r_{A-A}(d):=\vert \{(a,b)\in A\times A\vert a-b=d\}\vert \ge m\). It is also shown that this construction is optimal, proving that, for any convex set with cardinality \(n\) and any \(d\ne 0\), \(r_{A-A}(d)\le\lfloor n/2\rfloor\).