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Revisiting sums and products in countable and finite fields

Ioannis Kousek

TL;DR

The paper develops a polynomial ergodic framework for the affine group actions on countable fields, then translates these ergodic insights into density and partition results for sum–product patterns in both countable and finite fields. Using a polynomial van der Corput induction and a Furstenberg-type correspondence principle, it proves that large sets in characteristic zero contain many patterns of the form $\{x,p(x)+y,xy\}$, and it extends the analysis to finite fields via finitistic methods and a new colouring trick. It generalizes Shkredov’s results to all finite fields, establishes a double ergodic theorem for affine actions, and provides a conditional, elementary generalisation of Green–Sanders’ theorem on monochromatic sum–product patterns. Overall, the work connects ergodic theory, combinatorial number theory, and finite-field incidence to show that sum–product patterns are robust phenomena across both countable and finite settings, with implications for partition regularity and recurrence in dynamical contexts.

Abstract

We establish a polynomial ergodic theorem for actions of the affine group of a countable field $K$. As an application, we deduce--via a variant of Furstenberg's correspondence principle--that for fields of characteristic zero, any "large" set $E\subset K$ contains "many" patterns of the form $\{p(x)+y,xy\}$, for every non-constant polynomial $p(x)\in K[x]$. Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a new finitistic variant of Bergelson's "colouring trick", show that for $r\in \mathbb{N}$ fixed, any $r-$colouring of a large enough finite field will contain monochromatic patterns of the form $\{x,p(x)+y,xy\}$. In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalisation of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic "colouring trick", we provide a conditional, elementary generalisation of Green and Sanders' $\{x,y,x+y,xy\}$ theorem.

Revisiting sums and products in countable and finite fields

TL;DR

The paper develops a polynomial ergodic framework for the affine group actions on countable fields, then translates these ergodic insights into density and partition results for sum–product patterns in both countable and finite fields. Using a polynomial van der Corput induction and a Furstenberg-type correspondence principle, it proves that large sets in characteristic zero contain many patterns of the form , and it extends the analysis to finite fields via finitistic methods and a new colouring trick. It generalizes Shkredov’s results to all finite fields, establishes a double ergodic theorem for affine actions, and provides a conditional, elementary generalisation of Green–Sanders’ theorem on monochromatic sum–product patterns. Overall, the work connects ergodic theory, combinatorial number theory, and finite-field incidence to show that sum–product patterns are robust phenomena across both countable and finite settings, with implications for partition regularity and recurrence in dynamical contexts.

Abstract

We establish a polynomial ergodic theorem for actions of the affine group of a countable field . As an application, we deduce--via a variant of Furstenberg's correspondence principle--that for fields of characteristic zero, any "large" set contains "many" patterns of the form , for every non-constant polynomial . Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a new finitistic variant of Bergelson's "colouring trick", show that for fixed, any colouring of a large enough finite field will contain monochromatic patterns of the form . In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalisation of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic "colouring trick", we provide a conditional, elementary generalisation of Green and Sanders' theorem.
Paper Structure (12 sections, 37 theorems, 164 equations)

This paper contains 12 sections, 37 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Historic background
  3. Main results
  4. Preliminaries and some useful results
  5. The action of the affine group
  6. Ergodic theorems and van der Corput lemmas
  7. Proofs of Theorems \ref{['main theorem density version']} and \ref{['main ergodic theorem']}
  8. A finite fields version of Theorem \ref{['main theorem density version']}
  9. A new "colouring trick" and partition regularity for finite fields
  10. Proof of Theorem \ref{['main ergodic theorem 2']}
  11. Generalization of Shkredov's theorem
  12. A conditional generalisation of Green and Sanders' theorem

Key Result

Theorem 1.2

For any finite colouring of $\mathbb N$ there exist (infinitely many) $x,y\in \mathbb N$ such that $\{x,x+y,xy\}$ is monochromatic.

Theorems & Definitions (73)

  • Theorem 1.2: Moreira
  • Theorem 1.3: Shkredov
  • Theorem 1.4: Shkredov
  • Theorem 1.5: Green-Sanders
  • Theorem 1.6: Bergelson-Moreira
  • Definition 1.7
  • Theorem 1.8: Bergelson-Moreira
  • Theorem 1.9: Bergelson-Moreira
  • Remark
  • Definition 1.10
  • ...and 63 more