On Products of Shifts in Arbitrary Fields

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.