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 $\frac {99}{41} = 2.413\ldots$ improves slightly on the exponent of $\frac {17}{7} = 2.428\ldots$ due to Rudnev and Shkredov. Using this estimate we prove that the extension operator for the three dimensional paraboloid in prime order fields maps $L^2 \rightarrow L^{r}$ for $r >\frac {188}{53}=3.547\ldots$ improving the previous range of $r\geq \frac {32}{9}= 3.\overline {555}$.