# On Products of Shifts in Arbitrary Fields

## TL;DR

This paper proves new lower bounds on the size of product sets involving shifts in arbitrary fields, improving previous exponents by adapting techniques from additive combinatorics.

## Contribution

It introduces an adapted approach to establish sharper bounds on product sets in arbitrary fields, surpassing earlier incidence theorem results.

## Key findings

- |A(A+1)|   |A|^{11/9}
- |AA| + |(A+1)(A+1)|   |A|^{11/9}
- Improved bounds over previous results for finite fields with characteristic p.

## Abstract

We adapt the approach of Rudnev, Shakan, and Shkredov to prove that in an arbitrary field $\mathbb{F}$, for all $A \subset \mathbb{F}$ finite with $|A| < p^{1/4}$ if $p:= Char(\mathbb{F})$ is positive, we have $$|A(A+1)| \gtrsim |A|^{11/9}, \qquad |AA| + |(A+1)(A+1)| \gtrsim |A|^{11/9}.$$ This improves upon the exponent of $6/5$ given by an incidence theorem of Stevens and de Zeeuw.