Arithmetic and Geometric Progressions in Productsets over Finite Fields

Given two sets $\text{cA},\text{cB}\subseteq {\text{F}}_q$ of elements of the finite field ${\text{F}}_q$ of $q$ elements, we show that the productset $$\text{cA}\text{cB}=\{ab\mathrm{\mid }a\in \text{cA},b\in \text{cB}\}$$ contains an arithmetic progression of length $k\ge 3$ provided that $k<p$, where $p$ is the characteristic of ${\text{F}}_q$, and $# \cA # \cB \ge 3q^{2d-2/k}$. We also consider geometric progressions in a shifted productset $\text{cA}\text{cB}+h$, for $f\in {\text{F}}_q$, and obtain a similar result.