# Four-variable expanders over the prime fields

## TL;DR

This paper improves bounds on the number of distinct cubic distances in small sets over prime fields and explores new families of expanders in four and five variables, providing explicit exponents for related sum-product problems.

## Contribution

It presents improved lower bounds on cubic distances and introduces new families of expanders in multiple variables, along with an explicit exponent for a quadratic polynomial sum-product problem.

## Key findings

- |(A-A)^3+(A-A)^3|\u2265 c|A|^{8/7} for small sets A in prime fields
- Identifies new families of expanders in four and five variables
- Proves \,max\,|A+A|,|f(A,A)||A|^{6/5} for certain quadratic polynomials

## Abstract

Let $\mathbb{F}_p$ be a prime field of order $p>2$, and $A$ be a set in $\mathbb{F}_p$ with very small size in terms of $p$. In this note, we show that the number of distinct cubic distances determined by points in $A\times A$ satisfies \[|(A-A)^3+(A-A)^3|\gg |A|^{8/7},\] which improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables. We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove that \[\max \left\lbrace |A+A|, |f(A, A)|\right\rbrace\gg |A|^{6/5},\] where $f(x, y)$ is a quadratic polynomial in $\mathbb{F}_p[x, y]$ that is not of the form $g(\alpha x+\beta y)$ for some univariate polynomial $g$.