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

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.