TY - JOUR

T1 - Ergodic theorem involving additive and multiplicative groups of a field {x + y, xy} and patterns

AU - Bergelson, Vitaly

AU - Moreira, Joel

N1 - Funding Information:
The first author was supported by NSF grant DMS-1162073.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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 ∈ 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 and u + v ∈ E1 uv ∈ E2.

AB - 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 ∈ 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 and u + v ∈ E1 uv ∈ E2.

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U2 - 10.1017/etds.2015.68

DO - 10.1017/etds.2015.68

M3 - Article

AN - SCOPUS:84943768254

VL - 37

SP - 673

EP - 692

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 3

ER -