Additive energy and the Falconer distance problem in finite fields (Q2855606)

Let \(A\) and \(B\) be subsets of \(\mathbb F_q^d\), with \(d\geq 2\). Define NEWLINE\[NEWLINE \Delta (A, B)=\{ ||x-y||: x\in A, y\in B\}, NEWLINE\]NEWLINE where \(||z||=\sum z_i^2\). The finite field analog of the Falconer distance problem is to find the minimal value of \(|A||B|\) that implies \(|\Delta (A, B)|\gtrsim q\), where \(a\gtrsim b\) means there exists a constant \(C>0\), independent of \(q\), such that \(Ca\geq b\).NEWLINENEWLINEThe authors give two constructions of sets \(A, B\) such that \(|A||B|\sim q^d\) and \(|\Delta (A, B)|\gtrsim q\). The first requires \(d=2\). Let \(P(x,y)\in\mathbb F_q[x,y]\) have no linear factor and set \(V\) equal to its zero set. Then for \(B\subset V\), \(|B|\sim q\), and any \(A\) with \(|A|\sim q\) they show \(|A||B|\sim q^2\) and \(|\Delta (A, B)|\sim q\). The second construction is for any \(d\) and requires that \(B\) be a Salem set.