Ergodic Theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ 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 [11], showing that for any finite field $F$ and any subsets $E_1,E_2\subset F$ with $\mathrm{\mid }{E}_1\mathrm{\mid }\mathrm{\mid }{E}_2\mathrm{\mid }>6\mathrm{\mid }F\mathrm{\mid }$, there exist $u,v\in F$ such that $u+v\in E_1$ and $uv\in E_2$.