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 +y,xy\ . We also show that for any finite coloring of K there are many x,y∈ 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. Combinatorica 32 (5) (2012), 497–511], showing that for any finite field F and any subsets E₁,E₂⊂ F with |E₁|\,|E₂|>6|F| , there exist u,v∈ F such that u+v∈ E₁ and uv∈ E₂ .