Abstract
Let Fp be a prime field, and E a set in Fp2. Let Δ(E)={||x−y||:x,y∈E}, the distance set of E. In this paper, we provide a quantitative connection between the distance set Δ(E) and the set of rectangles determined by points in E. As a consequence, we obtain a new lower bound on the size of Δ(E) when E is not too large, improving a previous estimate due to Lund and Petridis (2018). We also study a variant of the distance problem for Cartesian product sets. More precisely, for not too big sets A⊂Fp, we will prove that |(A−A)2−(A−A)2|≳|A|32+1142.
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