On growth of the set $A(A+1)$ in arbitrary finite fields

TL;DR

This paper establishes a new explicit lower bound on the size of the set A(A+1) in finite fields, improving previous bounds and applying under more general conditions.

Contribution

It provides the first explicit lower bound for A(A+1) in finite fields under relaxed structural restrictions, advancing additive combinatorics.

Findings

Improved lower bounds on |A(A+1)| in finite fields

Applicable under more general structural conditions

Enhances understanding of product sets in finite fields

Abstract

Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is a power of a prime. For a set $A \subset \mathbb{F}_q$, under certain structural restrictions, we prove a new explicit lower bound on the size of the product set $A(A + 1)$. Our result improves on the previous best known bound due to Zhelezov and holds under more relaxed restrictions.