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

## TL;DR

This paper establishes a new upper bound on the number of rectangles with vertices in a small subset of a finite plane and applies this to improve the restriction estimate for the paraboloid in finite fields.

## Contribution

It introduces a tighter bound on rectangle counts in finite fields and enhances the restriction estimate for the paraboloid in three dimensions.

## Key findings

- Bound on rectangles: $O_\epsilon (|A|^{99/41+\epsilon})$
- Improved restriction estimate: $L^2
ightarrow L^{r}$ for $r > 188/53$
- Enhanced understanding of geometric configurations in finite fields

## 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 $r\geq \frac{32}{9}= 3.\overline{555}$.