Counting rectangles and an improved restriction estimate for the paraboloid in $F_p^3$

Abstract

Given $A\subset F_p^2$ a sufficiently small set in the plane over a prime residue field, we prove that there are at most $O_{\epsilon }(|A{|}^{\frac{99}{41}+\epsilon })$ rectangles with corners in $A$. The exponent $\frac{99}{41}=2.413\dots$ improves slightly on the exponent of $\frac{17}{7}=2.428\dots$ due to Rudnev and Shkredov. Using this estimate we prove that the extension operator for the three dimensional paraboloid in prime order fields maps $L^2\to L^r$ for $r>\frac{188}{53}=3.547\dots$ improving the previous range of $r\ge \frac{32}{9}=3.\overline{555}$.

8 pages, no figures. v2/v3: very minor edits