Ergodic theorem involving additive and multiplicative groups of a field and patterns

We establish a ‘diagonal’ ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg’s correspondence principle, prove that any ‘large’ set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular, we obtain an alternative proof for a result obtained by Cilleruelo [Combinatorial problems in finite fields and Sidon sets. Combinatorica32(5) (2012), 497–511], showing that for any finite field $F$ and any subsets $E_{1},E_{2}\subset F$ with $|E_{1}|\,|E_{2}|>6|F|$, there exist $u,v\in F$ such that $u+v\in E_{1}$ and $uv\in E_{2}$.