On the Erdős-Falconer distance problem for two sets of different size in vector spaces over finite fields (Q1955621)

Given a finite set \(E\) of vectors in \(\mathbb R^s\), the Erdős distance conjecture concerns bounding the set of lengths in \(E\) by the size of \(E\). \textit{A. Iosevich} and \textit{M. Rudnev} [Trans. Am. Math. Soc. 359, No. 12, 6127--6142 (2007; Zbl 1145.11083)] considered a finite field version using \[ |\boldsymbol \alpha|^2=\sum_{i=1}^s \alpha_i^2, \] for \(\pmb{ \alpha}\in\mathbb F_q^s\). Here the author considers two sets \(E, F\subset \mathbb F_q^s\). Set \[ \Delta (E, F)=\{ |\mathbf x -\mathbf y|^2 : \mathbf x\in E, \mathbf y\in F\}. \] A simple extension of Iosevich and Rudnev yields that if \((\# E)(\# F)\gg q^{s+1}\) then \(\#\Delta (E, F)\gg q\). This is improved to: if \[ \begin{aligned} (\# E)(\# F) &\gg (900+\log q)q^s\qquad\text{and}\\ \max\{ \# E, \# F\} &\gg q^{(s+1)/2}\log q \end{aligned} \] then \(\#\Delta (E, F)\gg q\). This is close to a conjecture of Koh and Shen which says that, for \(s\) even and a sufficiently large constant \(C\), if \((\# E)(\# F)\geq Cq^s\) then \(\#\Delta (E, F)\gg q\). Note that the conjecture is only for even \(s\) while the result here is valid for both even and odd \(s\).