Published online by Cambridge University Press: 06 October 2015
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
$uv\in E_{2}$.
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