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

Vitaly Bergelson, Joel Moreira

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)673-692
Number of pages20
JournalErgodic Theory and Dynamical Systems
Volume37
Issue number3
DOIs
StatePublished - May 1 2017

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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