Revisiting sums and products in countable and finite fields

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 $\{p(u)+v,uv\}$, 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 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 $\{u,p(u)+v,uv\}$. 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 generalization of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic ‘colouring trick’, we provide a conditional, elementary generalization of Green and Sanders’ $\{u,v,u+v,uv\}$ theorem.

1 Introduction

1.1 Historic background

A well-known and still open question of Hindman (see, for example, [9]) reads as follows.

Question 1.1. Given any finite colouring of $\mathbb N$ , do there always exist $u,v\in \mathbb N$ such that $\{u,v,u+v,uv\}$ is monochromatic, that is, $u,v,u+v$ and $uv$ all have the same colour?

In [11], Moreira proved the following result marking significant progress towards an answer to Question 1.1.

Theorem 1.2. [11, Corollary 1.5]

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

Prior to Moreira’s theorem, Shkredov [13] addressed its analogue for finite fields of prime order proving two density results.

Theorem 1.3. [13, Theorem 2.1]

Let $\mathbb Z_p$ be a finite field of prime order p. If ${B_1,B_2 \subset \mathbb Z_p}$ are any sets with $|B_1||B_2|\geq 20p$ , then there exist $u,v\in \mathbb Z^{*}_p:= \mathbb Z_p \setminus \{0\}$ such that ${u+v\in B_1}$ and $uv\in B_2$ .

Theorem 1.4. [13, Theorem 1.2]

Let $\mathbb Z_p$ be a finite field of prime order p. If $B_1,B_2, B_3 \subset \mathbb Z_p$ are any sets with $|B_1||B_2||B_3|\geq 40p^{5/2}$ , then there exist $u,v\in \mathbb Z^{*}_p$ such that $u+v\in B_1$ , $uv\in B_2$ and $u\in B_3$ .

It follows from Theorem 1.4 that if $\mathbb Z_p$ is r-coloured and p is large enough relative to r, then there exist $u,v\in \mathbb Z^{*}_p$ such that $\{u,u+v,uv\}$ is monochromatic. Later, the analogue of Question 1.1 for finite fields of prime order was solved by Green and Sanders in [7] via the following quantitative result.

Theorem 1.5. [7, Theorem 1.2]

Let $r\in \mathbb N$ be fixed and $\mathbb Z_p$ be a finite field of prime order p, with p large enough. For any r-colouring of $\mathbb Z_p$ , there are at least $c_r p^2$ monochromatic quadruples $\{u,v,u+v,uv\}$ , where $c_r>0$ does not depend on p.

Observe that Theorems 1.3 and 1.4 are density results, while there is no density version of the partition regularity Theorem 1.5. This was pointed out by Shkredov in [13].

In the context of countable fields, Bowen and Sabok in [4] gave a positive answer to the analogue of Question 1.1. By a compactness principle, they also solved the analogue of this question for all finite fields as a corollary of their main theorem.

Before that, Bergelson and Moreira in [3] established the following analogue of Theorem 1.2 using methods from ergodic theory.

Theorem 1.6. [3, Theorem 1.2]

Let K be a countable field and consider a finite colouring $K = C_1 \cup \cdots \cup C_r$ . Then, there exists a colour $C_i$ , $1\leq i \leq r$ , and ‘many’ $u, v \in K^{*}$ , such that $\{u,u+v,uv\} \subset C_i.$

In this setting, an appropriate notion of largeness, which guarantees patterns involving both addition and multiplication in any large set, turns out to be that of positive upper density with respect to double Følner sequences. We recall the definition given in [3].

Definition 1.7. Let K be a countable field. A double Følner sequence in K is a sequence of (non-empty) finite subsets $(F_N)_{N\in \mathbb N} \subset K$ that is asymptotically invariant under any fixed affine transformation of K, that is,

$$ \begin{align*}\lim_{N\to \infty} \frac{| F_N \cap (x+F_N) |}{|F_N|}=\lim_{N\to \infty} \frac{| F_N \cap (xF_N) |}{|F_N|}=1\end{align*} $$

for any $x\in K^{*}$ .

This notion of sequence allows us to define asymptotic densities with good properties such as shift invariance. For a countable field K and $(F_N)_{N\in \mathbb N}$ a double Følner sequence in K as above, given a set $E\subset K$ , its upper density with respect to $(F_N)_{N\in \mathbb N}$ is defined as

$$ \begin{align*}\overline{{\mathop{}\!\mathrm{d}}}_{(F_N)}(E)=\limsup_{N\to \infty} \frac{| E \cap F_N |}{|F_N|}.\end{align*} $$

Moreover, its lower density with respect to $(F_N)_{N\in \mathbb N}$ is defined as

$$ \begin{align*}\underline{{\mathop{}\!\mathrm{d}}}_{(F_N)}(E)=\liminf_{N\to \infty} \frac{| E \cap F_N |}{|F_N|}\end{align*} $$

and whenever the limit exists, we say that E has a density with respect to $(F_N)_{N\in \mathbb N}$ given by ${\mathop {}\!\mathrm {d}}_{(F_N)}(E)=\overline {{\mathop {}\!\mathrm {d}}}_{(F_N)}(E)=\underline {{\mathop {}\!\mathrm {d}}}_{(F_N)}(E).$

Using a ‘colouring trick’, Bergelson and Moreira were able to recover Theorem 1.6 from essentially the following theorem, which we state vaguely.

Theorem 1.8. [3, Theorem 1.5]

Let K be a countable field, $(F_N)_{N\in \mathbb N}$ be a double Følner sequence in K and $E \subset K$ with $\overline {{\mathop {}\!\mathrm {d}}}_{F_N}(E)>0$ . Then, there exist ‘many’ $u,v\in K$ such that $\{u+v,uv\} \subset E$ .

An advantage of the statement of Theorem 1.8, over that of Theorem 1.6, is that its form can be handled with ergodic theoretic tools and methods. This is a general principle, discovered by Furstenberg in his seminal proof of Szemerédi’s theorem (see [6]). There, he introduced a correspondence principle, which often allows one to translate a problem of finding patterns in large sets (subsets of the integers, of semi-groups, of fields, etc.) to a problem about recurrence in measure-preserving systems.

The following ergodic theorem from [3], whose proof uses the group of affine transformations of a field K, defined as $\mathcal {A}_K:=\{f:x \mapsto ux+v | \ u,v\in K, u\neq 0 \}$ , implies Theorem 1.8. We write $A_u$ for the map $x \mapsto x+u$ if $u\in K$ and $M_u$ for $x\mapsto ux$ , if $u\in K^{*}:=K \setminus \{0\}$ .

Theorem 1.9. [3, Theorem 1.4]

Let K be a countable field and $(F_N)_{N\in \mathbb N}$ be a double Følner sequence in K. Let $(X,\mathcal {X},\mu )$ be a probability space and let $(T_g)_{g\in \mathcal {A}_K}$ denote a measure-preserving action of $\mathcal {A}_K$ on X. Then, given any $B\in \mathcal {X}$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \mu( A_{-u}B \cap M_{1/u}B) \geq (\mu(B))^2.\end{align*} $$

Remark 1.10. The fact that $(T_g)_{g\in \mathcal {A}_K}$ acts on $(X,\mathcal {X},\mu )$ by measure-preserving transformations means that $(T_g)_{g\in \mathcal {A}_K}$ is a group action on X, so that $T_g \circ T_h = T_{g \circ h}$ for any $g,h\in \mathcal {A}_K$ , and that $\mu (B)=\mu (T_g^{-1}B)$ for any $B\in \mathcal {X}$ and $g\in \mathcal {A}_K$ . Also, in an abuse of notation, we write $A_u$ for $T_{A_u}$ and $M_u$ for $T_{M_u}$ , where $u\in K^{*}$ .

1.2 Main results

A question that occurs naturally is whether we can extend Theorem 1.6 by finding monochromatic patterns of the form $\{u,p(u)+v,uv\}$ , where $p(x)$ is a polynomial over K, other than $p(x)=x$ . This is addressed by our first main result (stated somewhat vaguely for now) which we formulate after an important—throughout this paper—definition.

Definition 1.11. Given a field K with prime characteristic $\text {char}(K)=q$ , we say that a non-constant polynomial $p(x)\in K[x]$ is admissible for K if $\deg (p(x))\leq q-1$ . If K is a countable field with $\text {char}(K)=0$ , then any non-constant polynomial $p(x)\in K[x]$ is admissible for K.

Theorem 1.12. Let K be a countable field and $p(x)\in K[x]$ be any admissible polynomial. Then, for any finite colouring $K=C_1 \cup \cdots \cup C_r,$ there exists a colour $C_j$ , $1\leq j \leq r$ , and ‘many’ $u,v \in K^{*}$ , so that $\{u,p(u)+v,uv\} \subset C_j.$

The density theorem that we will use to prove Theorem 1.12 is the following.

Theorem 1.13. Let K be a countable field, $(F_N)_{N\in \mathbb N}$ be a double Følner sequence in K and $E \subset K$ with $\overline {{\mathop {}\!\mathrm {d}}}_{(F_N)}(E)>0$ . Then, for any admissible polynomial $p(x) \in K[x]$ , there exist ‘many’ $u,v\in K$ such that $\{p(u)+v,uv\} \subset E$ .

In the same spirit as in the end of § 1.1, Theorem 1.13 is implied by an ergodic theorem.

Theorem 1.14. Let K, $p(x) \in K[x]$ and $(F_N)_{N\in \mathbb N}$ be as in the statement of Theorem 1.13. Let $(X,\mathcal {X},\mu )$ be a probability space and let $(T_g)_{g\in \mathcal {A}_K}$ denote a measure-preserving action of $\mathcal {A}_K$ on X. Then, given any $f\in L^2(X,\mu )$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} M_{u}A_{-p(u)}f = Pf,\end{align*} $$

where the limit is in $L^2$ and $P:L^2(X,\mu ) \to L^2(X,\mu )$ denotes the orthogonal projection onto the subspace of $\mathcal {A}_K$ -invariant functions.

The proof of this statement is based on that of Bergelson and Moreira’s proof of Theorem 1.9, with additional applications of van der Corput type of lemmas to facilitate an induction argument on the degree of the polynomial. This appears especially in the proof of the polynomial mean ergodic theorem of Proposition 3.3 below.

We also finitize the arguments used to prove Theorem 1.14 to recover the following analogue of our main density result, Theorem 1.13, in the setting of finite fields.

Theorem 1.15. Let F be a finite field and let $p(x)\in F[x]$ be an admissible polynomial over F of degree $q:=\deg (p(x))$ . Then, if $B_1,B_2 \subset F$ with $|B_1||B_2|> 2(q+2)|F|^{2-(1/2^{q-1})}$ , there are $u,v\in F^{*}$ , so that $uv\in B_1$ and $p(u)+v \in B_2$ .

In particular, letting $B=B_1=B_2$ , we have the finite field version of the density statement that there exist $u,v\in F^{*}$ such that $\{p(u)+v,uv\} \subset B$ , provided $B \subset F$ is large enough.

We also produce a finitistic version of the ‘colouring trick’ mentioned earlier and, with the aid of Theorem 1.15, recover the next partition regularity result.

Theorem 1.16. Let $r,q\in \mathbb N$ be fixed. Then, there exists $n(r,q)\in \mathbb N$ with the following property. If F is any finite field with $|F|\geq n(r,q)$ and $\mathrm {char}(F)>q$ , and $p(x)\in F[x]$ is a polynomial of $\deg (p(x))=q$ , then for any finite colouring $F=C_1 \cup \cdots \cup C_r$ , there is a colour $C_j$ and $u,v\in F^{*}$ , such that $\{u,p(u)+v,uv\} \subset C_j.$

Remark 1.17. The assumption $\text {char}(F)>q$ is only to ensure that the polynomial ${p(x) \in F[x]}$ is admissible according to Definition 1.11.

A special case of this theorem (when $p(x)=x$ ) is the partition regularity corollary of Shkredov’s Theorem 1.4 mentioned after its statement. An advantage of the ergodic theoretic techniques used here is that we can recover more general polynomial patterns and also that the result holds for all finite fields and not only $\mathbb Z_p$ . A perhaps more interesting feature, however, is the use of a ‘colouring trick’ in the finitistic setting, which, in a way, allows us to recover this partition regularity statement from a weaker density theorem. Noteworthily, a similar ‘colouring trick’ has appeared before in the literature (see [12, Proposition $2.2$ ]).

In a different direction, we are also interested in the question of [3, §6.4]. Namely, is it true that under the assumptions of Theorem 1.9 above, we get triple intersections of the form $\mu (B \cap A_{-u}B \cap M_{1/u}B )>0$ for some $u \in K^{*}$ ? A generalization of the next non-commutative double ergodic theorem, without the assumption of ergodicity, would answer this question in the affirmative.

Theorem 1.18. Let K be a countable field and $(F_N)_{N\in \mathbb N}$ be a double Følner sequence in K. Let $(X,\mathcal {X},\mu )$ be a probability space, let $(T_g)_{g\in \mathcal {A}_K}$ denote a measure-preserving action of $\mathcal {A}_K$ on X and (crucially) we further assume that the action of the additive subgroup $S_A=\{A_u: u\in K\}$ is ergodic. (The action $(T_g)_{g\in G}$ of a group G on a probability space $(X,\mathcal {X},\mu )$ is ergodic if for any $B\in \mathcal {X}$ , we have that $T_gB=B\ \text {for all} g\in G {\implies} \mu (B)\in \{0,1\}$ .) Then, given any $B \in \mathcal {X}$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \mu(B \cap A_{-u}B \cap M_{1/u}B ) \geq (\mu(B))^3.\end{align*} $$

Unfortunately, we were unable to recover the result in its full generality. However, we make a natural conjecture.

Conjecture 1.19. In the context of Theorem 1.18, if $S_A$ does not act ergodically, then given any $B \in \mathcal {X}$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \mu(B \cap A_{-u}B \cap M_{1/u}B ) \geq (\mu(B))^4.\end{align*} $$

In a relevant direction, Theorem 1.3 was generalized to all finite fields, initially by Cilleruelo [5, Corollary $4.2$ ] and subsequently by Hanson [8, Theorem $1$ ] and Bergelson, Moreira [3, Theorem $5.3$ ]. However, a generalization of Theorem 1.4 to all finite fields remained open and we address this problem hereby through a ‘finitization’ of Theorem 1.18.

Theorem 1.20. Let F be any finite field and let $B_1,B_2,B_3 \subset F$ be any sets satisfying $|B_1||B_2||B_3|\geq 8|F|^{5/2}$ . Then, there exist $u,v\in F^{*}$ such that $u+v\in B_1$ , $uv\in B_2$ and ${u\in B_3}$ .

The ideas and techniques appearing in the proof of Theorem 1.18 spring from classical ergodic theoretic arguments used in proving multiple ergodic theorems. In this regard, the proof of Theorem 1.20, which is more or less a ‘finitization’ of the above-mentioned proof, is different from Shkredov’s original combinatorial proof of Theorem 1.4.

Finally, by using the finitistic ‘colouring trick’ and a finitistic version of Conjecture 1.19, we provide an elementary, conditional proof of the following generalization of Green and Sanders’ Theorem 1.5.

Conjecture 1.21. Let $r\in \mathbb N$ be fixed. Then, there is $n(r)\in \mathbb N$ , such that if F is any finite field with $|F|\geq n(r)$ and $F=C_1\cup \cdots \cup C_r$ , there are $c_r|F|^2$ monochromatic quadruples $\{u,v,u+v,uv\}$ , where $c_r>0$ does not depend on $|F|$ .

2 Preliminaries and some useful results

2.1 The action of the affine group

For a countable field K, we denote by $\mathcal {A}_K= \{f:x \mapsto ux+v:\ u,v\in K,\ u\neq 0\}$ the group of affine transformations of K, with the operation of composition. The additive subgroup of $\mathcal {A}_K$ is denoted by $S_A$ and consists of the transformations $A_u: x \mapsto x+u$ for $u\in K$ . Similarly, the multiplicative subgroup, denoted by $S_M$ , consists of transformations of the form $M_u: x \mapsto ux$ for $u\in K^{*}$ . The map $x \mapsto ux+v$ can be represented by the composition $A_vM_u$ and we have the trivial, but very useful throughout this paper, identity:

(2.1) $$ \begin{align} M_uA_v=A_{uv}M_u. \end{align} $$

The affine group appears naturally in our considerations because, for example, to find patterns $\{u+v,uv\}$ in a subset $E \subset K$ , we can show that for some $u\in K^{*}$ , the intersection $A_{-u}E \cap M_{1/u}E$ is non-empty.

We have already mentioned the utility of double Følner sequences as averaging schemes in K. The existence of such sequences was proved in [3, Proposition 2.4].

Proposition 2.1. Any countable field K admits a sequence of non-empty finite sets $(F_N)_{N\in \mathbb N}$ , which forms a Følner sequence for both the actions of the additive group $(K,+)$ and the multiplicative group $(K^{*},\cdot )$ . In other words, for any $u\in K^{*}$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{| F_N \cap (u+F_N) |}{|F_N|}=\lim_{N\to \infty} \frac{| F_N \cap (uF_N) |}{|F_N|}=1.\end{align*} $$

According to [3, Lemma 2.6], some transformations of double Følner sequences remain double Følner sequences.

Lemma 2.2. Let K be a countable field. If $(F_N)_{N\in \mathbb N}$ is a double Følner sequence in K and $b \in K^{*}$ , then $(bF_N)_{N\in \mathbb N}$ is still a double Følner sequence in K.

We will further consider a probability space $(X,\mathcal {X},\mu )$ and a measure-preserving action $(T_g)_{g\in \mathcal {A}_K}$ of $\mathcal {A}_K$ on X. In this context, we denote $L^2(X,\mu )$ by H and let $(U_g)_{g\in \mathcal {A}_K}$ be given by $(U_gf)(x)=f(T_g^{-1}x)$ for $x\in X$ and $f\in H$ . This is known as the unitary Koopman representation of $\mathcal {A}_K$ . Abusing notation, we will usually write $A_uf$ instead of $U_{A_u}f$ and $M_uf$ instead of $U_{M_u}f$ . By $P_A$ , we denote the orthogonal projection from H onto the subspace of vectors that are fixed by the action of the additive subgroup $S_A$ . Also, by $P_M$ , we denote the orthogonal projection from H onto the subspace of vectors fixed under the action of $S_M$ .

The useful and unintuitive fact that the projections $P_A$ and $P_M$ commute was established in of [3, Lemma 3.1].

Lemma 2.3. For any $f\in H$ , we have that

$$ \begin{align*}P_A P_Mf = P_M P_A f.\end{align*} $$

By Lemma 2.3, we see that $P_AP_Mf$ is invariant under the actions of both $S_A$ and $S_M$ , and that $P_AP_Mf$ is an orthogonal projection. Since the subgroups $S_A$ and $S_M$ generate the whole group $\mathcal {A}_K$ , it follows that $P=P_AP_M=P_MP_A$ is the orthogonal projection from H onto the subspace of vectors fixed under the action of $\mathcal {A}_K$ .

2.2 Ergodic theorems and van der Corput lemmas

The mean ergodic theorem for unitary representations of countable abelian groups, which we will extend later for our purposes, has the following form and a proof of this version can be found, for example, in [1, Theorem 5.4].

Theorem 2.4. Let G be a countable abelian group and $(F_N)_{N\in \mathbb N}$ be a Følner sequence in G. Let also H be a Hilbert space and $(U_g)_{g\in G}$ be a unitary representation of G on H. Then, for any $f\in H$ ,

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{g\in F_N} U_gf = Pf,\end{align*} $$

where the limit is in the strong topology of H and P denotes the orthogonal projection onto the subspace of vectors fixed under G.

Remark 2.5. One may consider, for example, the cases where, provided that $\mathcal {A}_K$ acts by measure-preserving transformations (m.p.t. for short) on a probability space $(X,\mathcal {X},\mu )$ , we have that $H=L^2(X,\mu )$ , $G=S_A$ or $G=S_M$ , and then $P=P_A$ or $P=P_M$ , respectively.

We will consider an adaptation of the van der Corput lemma for unitary representations of countable abelian groups. A proof—of a stronger version—appears in [2, Theorem 2.12].

Lemma 2.6. Let $(G,\cdot )$ be a countable abelian group and $(a_u)_{u\in G}$ be a bounded sequence of vectors in a Hilbert space H, indexed by the elements of G. Let $(F_N)_{N\in \mathbb N}$ be a Følner sequence in G. If

$$ \begin{align*}\lim_{M\to \infty} \frac{1}{|F_M|} \sum_{v\in F_M} \limsup_{N\to \infty} \frac{1}{|F_N|} \bigg|\!\sum_{u\in F_N} \langle a_{u \cdot v}, a_u\rangle \bigg| =0,\end{align*} $$

then also

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} a_u =0. \end{align*} $$

Remark 2.7. This, in particular, holds when $(G,\cdot )=(K,+)$ or when $(G,\cdot )=(K^{*},\cdot )$ for some countable field K and $(F_N)_{N\in \mathbb N}$ is a double Følner sequence in K.

Another version of the van der Corput lemma, which will be used in §6, follows as a corollary of the inequality given in Host and Kra’s book [10, Lemma 1, Ch. 21].

Proposition 2.8. Let $(G,\cdot )$ be a countable abelian group with identity $1$ and for each $b\in G$ , let $(a_u(b))_{u\in G}$ be a bounded sequence of vectors in a Hilbert space H with norm $\Vert \cdot \Vert $ , indexed by the elements of G. Let $(F_N)_{N\in \mathbb N}$ be a Følner sequence in G. If for all $d\neq 1$ ,

$$ \begin{align*} \lim_{M\to \infty} \frac{1}{|F_M|} \sum_{b\in F_M} \limsup_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \langle a_{u \cdot d}(b), a_u(b)\rangle =0,\end{align*} $$

then also

$$ \begin{align*}\lim_{M\to \infty} \frac{1}{|F_M|} \sum_{b\in F_M} \limsup_{N\to \infty} \bigg\Vert \frac{1}{|F_N|} \sum_{u\in F_N} a_u(b)\bigg\Vert^2=0. \end{align*} $$

For finite groups, a version of the van der Corput lemma is given by the following simple equality. We will use this to adapt our infinite ergodic theorems to the setting of finite fields.

Proposition 2.9. Let $(G,\cdot )$ be a finite group and $(f(g))_{g\in G}$ be a sequence taking values in a Hilbert space H. Then,

$$ \begin{align*}\bigg\Vert\!\sum_{g\in G} f(g)\bigg\Vert^2 = \sum_{g\in G} \sum_{h\in G} \langle f(g \cdot h), f(g)\rangle.\end{align*} $$

Finally, we shall find the next classical result useful.

Lemma 2.10. Let $(a_u)_{u\in G}$ be a bounded, non-negative sequence, indexed by elements of a countable (amenable) group G and let $(G_N)_{N\in \mathbb N}$ be a Følner sequence in G. Then,

$$ \begin{align*} \lim_{N\to \infty} \frac{1}{|G_N|} \sum_{u\in G_N} a_u = 0 \iff \lim_{N\to \infty} \frac{1}{|G_N|} \sum_{u\in G_N} a_u^2 = 0.\end{align*} $$

3 Proofs of Theorems 1.13 and 1.14

Throughout this section, we assume that K is a countable field, $(F_N)_{N\in \mathbb N}$ is a double Følner sequence in K and $p(x) \in K[x]$ is a non-constant admissible polynomial over K, according to Definition 1.11. We also let $(X,\mathcal {X},\mu )$ be a probability space on which we assume that $(T_g)_{g\in \mathcal {A}_K}$ acts by measure-preserving transformations. In consistency with the notation from §2, $H=L^2(X,\mu )$ , $P: H \to H$ denotes the orthogonal projection from H onto the subspace of functions fixed under the action of $\mathcal {A}_K$ , and $P_A$ , $P_M$ are the orthogonal projections on the subspaces of vectors fixed under the additive action $S_A$ and the multiplicative action $S_M$ , respectively. Moreover, $(U_g)_{g\in \mathcal {A}_K}$ is the unitary Koopman representation of $\mathcal {A}_K$ (for details, recall the discussion after Lemma 2.2). Again, for simplicity, we will write $A_u$ instead of $U_{A_u}$ and $M_u$ instead of $U_{M_u}$ .

Before embarking on the proof of Theorem 1.14, we show the ensuing, straightforward corollary of it.

Corollary 3.1. If K, $p(x)\in K[x]$ , $(F_N)_{N\in \mathbb N}$ and $(X,\mathcal {X},\mu )$ are as above, then for any $B\in \mathcal {X}$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \mu( A_{-p(u)}B \cap M_{1/u}B ) \geq (\mu(B))^2.\end{align*} $$

Proof. For $B\in \mathcal {X}$ , we see that

which can be written as (using that $M_u$ preserves $\mu $ for all $u\in K^{*}$ )

(3.1)

By Theorem 1.14 applied for , (3.1) becomes

For the last inequality, observe that P is an orthogonal projection and so

by the Cauchy–Schwarz inequality. Finally, because $P1=1$ , we have that

and thus, we conclude.

Remark 3.2. A similar argument shows that if in the context of Theorem 1.14 the action of $\mathcal {A}_K$ is also ergodic, then for any $B,C\in \mathcal {X}$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \mu( A_{-p(u)}B \cap M_{1/u}C ) \geq \mu(B) \mu(C).\end{align*} $$

For the special case $p(x)=x$ , the proof of Theorem 1.14 was given in [3]. We only mention that in the proof of the linear case in [3], the authors relied on a version of the mean ergodic Theorem 2.4 for the action of $S_A$ . For the polynomial case of Theorem 1.14, we will use the subsequent generalization, which is a polynomial mean ergodic theorem for the action of $S_A$ . For that, we will need an application of the van der Corput trick using the additive structure of K, which facilitates an induction argument on the polynomial’s degree.

Theorem 3.3. Let K be a countable field and $p(x) \in K[x]$ be admissible. Let also $(F_N)_{N\in \mathbb N}$ be a double Følner sequence in K and $(X,\mathcal {X},\mu )$ a probability space, on which $(T_{A_u})_{u\in K}$ acts by measure-preserving transformations (see also the beginning of this section). Then, given any $f\in H$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} A_{p(u)}f = P_Af,\end{align*} $$

where the limit is in the strong topology of H.

Proof. We prove the case $\text {char}(K)=q$ for some (see also Remark 3.4). If ${p(x)=ax+b}$ , where $a,b\in K$ and $a\neq 0$ , then it follows by the mean ergodic theorem that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|}\sum_{u\in F_N} A_{au+b}f = \lim_{N\to \infty} \frac{1}{|F_N|}\sum_{u\in aF_N+b} A_{u}f =P_Af.\end{align*} $$

Note that here, we used the fact that $(aF_N+b)_{N\in \mathbb N}$ is still a Følner sequence for the additive group $(K,+)$ , in view of Lemma 2.2 and the obvious observation that shifts of Følner sequences are also Følner sequences in any group. Now, assume the statement holds for polynomials of degree $m-1$ , where $2 \leq m \leq q-1$ , and let $p(x) \in K[x]$ have degree m, that is, $p(x)=q_0+q_1x+\cdots +q_mx^m$ , $q_0,\ldots ,q_m \in K$ and $q_m \neq 0$ . First, we let $f\in H$ be such that $P_Af=0$ and set $a_u=A_{p(u)}f, u \in K$ . Then, for any $b\in K^{*}$ , we have that

$$ \begin{align*}\langle a_{u+b}, a_u \rangle=\langle A_{p(u+b)-p(u)}f , f \rangle.\end{align*} $$

Observe that

$$ \begin{align*}p(u+b)-p(u)= q_m \sum_{k=0}^{m-1} \binom{m}{k} u^k \cdot b^{m-k} +r_b(u),\end{align*} $$

where $\deg (r_b(x)) \leq m-2$ . Therefore,

$$ \begin{align*}p(u+b)-p(u)= m \cdot (q_m b) u^{m-1} +r^{\prime}_b(u),\end{align*} $$

where $\deg (r_b'(x)) \leq m-2$ , and since $q_mb \neq 0$ , the above argument shows that the polynomial $g_{b}(x)=p(x+b)-p(x)$ has degree $m-1$ in $K[x]$ .

We note that an issue arises in allowing the polynomial’s degree to be q, in which case if, for example, $p(x)=x^q$ , then $g_{b}(x)=b^q$ is a constant, because $(x+b)^q=x^q+b^q$ in a field of characteristic q.

Returning to the proof, by the induction hypothesis and the assumption on f, we see that for any $b\neq 0$ ,

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \langle a_{u+b} , a_u \rangle=\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \langle A_{g_{b}(u)}f , f \rangle = \langle P_Af,f \rangle =0.\end{align*} $$

Thus, an application of the van der Corput trick as in Lemma 2.6 gives us that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} A_{p(u)}f =0,\end{align*} $$

in H, when $P_Af=0$ . Finally, for a general $f\in H$ , we can write $f=P_Af+(f-P_Af)$ and from the above and linearity, it follows that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} A_{p(u)}f = \lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} A_{p(u)}P_Af =P_Af.\\[-49pt] \end{align*} $$

Remark 3.4. Note that the same proof in the case of $\text {char(K)}=0$ (for example, when $K=\mathbb Q$ ) gives the same result for polynomials of arbitrarily large degree, because then, it always holds that $x \mapsto p(x+b)-p(x)$ is a polynomial of degree equal to $\deg (p(x))-1$ , when $b\neq 0$ .

We will now give the proof of Theorem 1.14, the statement of which we recall for the reader’s convenience.

Theorem 1.13. Let K, $(F_N)_{N\in \mathbb N}$ , $p(x) \in K[x]$ , $(X,\mathcal {X},\mu )$ and $(T_g)_{g\in \mathcal {A}_K}$ be as in the beginning of this section. Then, given any $f\in H$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} M_{u}A_{-p(u)}f = Pf,\end{align*} $$

where the limit is in the strong topology of H.

Proof. Let $f\in H$ and assume that $P_Af=0$ . For $u\in K^{*}$ , we now set $a_u=M_uA_{-p(u)}f$ and then, for any $b\in K^{*}$ , we have that

$$ \begin{align*} \langle a_{ub} , a_u \rangle = \langle A_{-p(ub)+p(u)/b}f , M_{1/b}f \rangle. \end{align*} $$

If $p(x)=q_0+q_1x+\cdots +q_mx^m$ , $q_0,\ldots ,q_m \in K$ and $q_m \neq 0$ ( $m<q$ if ${\text {char}(K)=q}$ ), then

$$ \begin{align*}p(ub)-p(u)/b=q_0\frac{b-1}{b}+u\bigg(q_1\frac{b^2-1}{b}\bigg)+\cdots+u^m\bigg(q_m\frac{b^{m+1}-1}{b} \bigg),\end{align*} $$

which, for $b\notin \{0,1,-1\}$ fixed, is also a polynomial of degree m. Thus, applying Theorem 3.3, we have that for $b\notin \{0,1,-1\}$ ,

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \langle a_{ub} , a_u \rangle = \langle P_Af , M_{1/b}f\rangle=0.\end{align*} $$

Once again, the van der Corput lemma implies that for $P_Af=0$ ,

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} M_uA_{-p(u)}f = 0,\end{align*} $$

and this allows us to conclude just like in the case of Theorem 3.3, after decomposing a general $f\in H$ as $f=P_Af+(f-P_Af)$ .

Using some quantitative bounds for the set of return times, which can be extracted from the proof of Corollary 3.1, and the variant of Furstenberg’s correspondence principle established in [3, Theorem 2.8], we can recover the following precise version of Theorem 1.13. The proof is a straightforward adaptation of the proof of [3, Theorem 2.5 from Theorem 2.10], which amounts to the special case that $p(x)=x$ .

Theorem 3.5. Let K be a countable field, $p(x) \in K[x]$ an admissible polynomial and $(F_N)_{N\in \mathbb N}$ be a double Følner sequence in K. Let $E \subset K$ with $\overline {{\mathop {}\!\mathrm {d}}}_{(F_N)}(E)>0.$ Then, for any $\epsilon>0$ , we have that

$$ \begin{align*}\underline{{\mathop{}\!\mathrm{d}}}_{(F_N)}( \{u\in K^{*}: \overline{{\mathop{}\!\mathrm{d}}}_{(F_N)}( (E-p(u)) \cap (E/u) ) \geq (\overline{{\mathop{}\!\mathrm{d}}}_{(F_N)}(E))^2-\epsilon \} )>0.\end{align*} $$

In less precise terms, for each element of a large set of $u \in K^{*}$ , there is a large set of $v\in K^{*}$ satisfying $\{ v+p(u),vu \} \subset E$ .

It may be instructive to compare this theorem with similar results in the setting of the integers. The so-called Furstenberg–Sárközy theorem, in particular, asserts (for example, although it is more general) that given any $p(x)\in \mathbb Z[x]$ with $p(0)=0$ , any set $E\subset \mathbb Z$ of positive upper density contains infinitely many pairs of the form $\{u,u+p(v)\}$ . However, for the above, where we are looking for patterns involving both addition and multiplication, we are relying on the structure of fields and restricting to densities that are both additively and multiplicatively invariant. This allows us to take advantage of useful ergodic techniques, particularly via the consideration of the field’s affine action, which lead to Theorem 1.14.

To conclude the results of this section, we give a precise statement of Theorem 1.12.

Theorem 3.6. Let K be a countable field, $(F_N)_{N\in \mathbb N}$ a double Følner sequence in K and $p(x)\in K[x]$ an admissible polynomial. Then, for any finite colouring ${K=C_1 \cup \cdots \cup C_r}$ , there exists a colour $C_j$ such that

$$ \begin{align*}\overline{{\mathop{}\!\mathrm{d}}}_{(F_N)}( \{u\in C_j: \overline{{\mathop{}\!\mathrm{d}}}_{(F_N)}( \{ v\in K: \{u,p(u)+v,uv\} \subset C_j \} ) \} )>0.\end{align*} $$

The proof of Theorem 3.6 is based on the ‘colouring trick’ from (and is almost identical to) the proof of [3, Theorem 4.1], and therefore is omitted. The only difference being that we rely on Corollary 3.1, while in [3], the authors relied on its special case of a linear polynomial.

It seems like our methods are not rigid enough to deal with non-admissible polynomials according to Definition 1.11 because of the comment in the proof of Theorem 3.3, so we make the following natural questions.

Question 3.7. Does Corollary 3.1 hold if $p(x) \in K[x]$ is not admissible?

Question 3.8. Does Theorem 3.6 (or a vague version as in Theorem 1.12) hold for non-admissible polynomials $p(x) \in K[x]$ ?

We note that a positive answer to Question 3.7 would also imply a positive answer to Question 3.8 by the same argument that is used for the case of admissible polynomials.

4 A finite fields version of Theorem 1.13

In this section, we will adapt the proof of Theorem 1.13 to the finite fields setting and prove Theorem 1.15.

For a finite field F, we consider its group of affine transformations, $\mathcal {A}_{F}$ , which consists of the maps of the form $x \mapsto ux+v,$ where $u\in F^{*}$ and $v\in F$ . We also let $(X,\mathcal {X},\mu )$ be a probability space on which $\mathcal {A}_{F}$ acts by measure-preserving transformations, with $(T_g)_{g\in \mathcal {A}_{F}}$ denoting the action. As before, we let $S_A=\{A_u : u\in F\}$ , where $A_u(x)=x+u$ and $S_M=\{M_u: u\in F^{*}\}$ , where $M_u(x)=xu$ . Also, in an abuse of notation, if $(U_g)_{g\in \mathcal {A}_{F}}$ is the Koopman representation of $\mathcal {A}_{F}$ on $L^2(X,\mu )$ , we write $A_u$ for $U_{A_u}$ and $M_u$ for $U_{M_u}$ , where, for example, for $f\in L^2(X,\mu )$ , we have that $U_{A_u}f(x)=f(T_{A_u}^{-1}x)=f(T_{A_{-u}}x)$ .

Moreover, if $P_A$ is the orthogonal projection onto the space of functions invariant under the subgroup $S_A$ , we see that $P_Af(x)=({1}/{|F|}) \sum _{u\in F} A_uf(x)$ and if $P_M$ is the projection onto the space of functions invariant under $S_M$ , then $P_Mf(x)=({1}/{|F^{*}|}) \sum _{u\in F^{*}} M_uf(x)$ . We will begin with a finitistic version of the polynomial mean ergodic theorem of §3 and then prove an analogue of Theorem 1.14. As in the infinite case, $P_A$ and $P_M$ exhibit commuting behaviour (see the proof of [3, Theorem 5.1]).

Proposition 4.1. For $f\in L^2(X,\mu )$ and $P_A$ , $P_M$ as above, we have that $P_AP_Mf=P_MP_Af$ .

Thus, $P_AP_M$ is an orthogonal projection onto the subspace of functions invariant under $\mathcal {A}_F$ . The promised finitistic analogue of Theorem 3.3 is this.

Proposition 4.2. Let F be a finite field and assume that $\mathcal {A}_{F}$ acts on $(X,\mathcal {X},\mu )$ as in the beginning of this section. Let also $p(x) \in F[x]$ be an admissible polynomial of degree $q:=\deg (p(x))$ . Then, for any $f\in L^2(X,\mu )$ , we have that

$$ \begin{align*}\bigg\Vert \frac{1}{|F|} \sum_{u\in F} A_{p(u)}f - P_Af\bigg\Vert_2^2 \leq \frac{q-1}{|F|^{1/2^{q-2}}}\Vert f-P_Af\Vert_2^2.\end{align*} $$

Proof. If $p(x)=ax+b$ , $a,b\in F$ and $a\neq 0$ , this is obvious, for $p(F)=\{au+b: u\in F\}=F$ , whence it is enough to make a change of variables and use the definition of $P_A$ . Assume now that the conclusion holds for polynomials of degree at most $q<r-1$ and let $p(x) \in F[x]$ be a polynomial of degree $q+1\leq r-1$ , where $\text {char}(F)=r$ for some . Assume first that $f\in L^2(X,\mu )$ is such that $P_Af=0$ . By Proposition 2.9, it follows that

(4.1) $$ \begin{align} \bigg\Vert \frac{1}{|F|} \sum_{u\in F} A_{p(u)}f\bigg\Vert_2^2 = \frac{1}{|F|} \sum_{v\in F} \frac{1}{|F|} \sum_{u\in F} \langle A_{p(u+v)-p(u)}f ,f \rangle. \end{align} $$

Since $\deg (p(x))=q+1\leq r-1$ , the polynomial $p(x+v)-p(x)$ has degree q for any $v\neq 0$ (this would no longer be true if the degree of $p(x)$ was r just like the infinite field case), and since $P_Af=0$ , the induction hypothesis implies that

(4.2) $$ \begin{align} \bigg\Vert \frac{1}{|F|} \sum_{u\in F} A_{p(u+v)-p(u)}f\bigg\Vert_2^2 \leq \frac{q-1}{|F|^{1/2^{q-2}}}\Vert f\Vert_2^2. \end{align} $$

Also, notice that

$$ \begin{align*}\frac{1}{|F|} \!\sum_{v\in F} \frac{1}{|F|} \!\sum_{u\in F} \langle A_{p(u+v)-p(u)}f ,f \rangle \leq \frac{1}{|F|}\Vert f\Vert_2^2 + \frac{1}{|F|} \!\sum_{v\in F^{*}} \frac{1}{|F|} \sum_{u\in F} \langle A_{p(u+v)-p(u)}f ,f \rangle,\end{align*} $$

which, by an application of the Cauchy–Schwarz inequality, is bounded above by

(4.3) $$ \begin{align} \frac{1}{|F|}\Vert f\Vert_2^2 + \max_{v\in F*} \bigg\Vert \frac{1}{|F|} \sum_{u\in F} A_{p(u+v)-p(u)}f\bigg\Vert_2\Vert f\Vert_2. \end{align} $$

Using (4.2) in (4.3) and then by (4.1), it follows that

(4.4) $$ \begin{align} \bigg\Vert \frac{1}{|F|} \sum_{u\in F} A_{p(u)}f\bigg\Vert_2^2 \leq \frac{1}{|F|} \Vert f\Vert_2^2 + \frac{\sqrt{q-1}}{|F|^{1/2^{q-1}}}\Vert f\Vert_2^2 \leq \frac{q}{|F|^{1/2^{q-1}}}\Vert f\Vert_2^2. \end{align} $$

Finally, for general $f\in L^2(X,\mu )$ , we have that

$$ \begin{align*}\frac{1}{|F|} \sum_{u\in F} A_{p(u)}f =\frac{1}{|F|} \sum_{u\in F} A_{p(u)}P_Af + \frac{1}{|F|} \sum_{u\in F} A_{p(u)}\tilde{f},\end{align*} $$

where $\tilde {f}=f-P_Af$ , so that $P_A\tilde {f}=0.$ Clearly, $({1}/{|F|}) \sum _{u\in F} A_{p(u)}P_Af = P_Af$ , and combining this with (4.4), the result follows.

We isolate the following estimate that appears in the proof of the finitistic analogue of Corollary 3.1, that is, Theorem 4.4 below. This estimate is the finitistic analogue of Theorem 1.14 for functions orthogonal to the space of functions fixed under the action of $S_A$ .

Proposition 4.3. Let F be a finite field and assume that $\mathcal {A}_F$ acts on $(X,\mathcal {X},\mu )$ as in the beginning of this section. Let also $p(x) \in F[x]$ be an admissible polynomial of degree $q:=\deg (p(x))$ . Let for some $C\in \mathcal {X}$ . Then,

(4.5) $$ \begin{align} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-p(u)}f\bigg\Vert_2^2 < 2(q+2)\mu(C)/|F^{*}|^{1/2^{q-1}}. \end{align} $$

Proof. From Proposition 2.9, we have that

(4.6) $$ \begin{align} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-p(u)}f\bigg\Vert_2^2 &= \frac{1}{|F^{*}|} \sum_{u\in F^{*}} \frac{1}{|F^{*}|} \sum_{v\in F^{*}} \langle M_{uv}A_{-p(uv)}f , M_uA_{-p(u)f} \rangle\nonumber\\ &= \frac{1}{|F^{*}|} \sum_{v\in F^{*}} \frac{1}{|F^{*}|} \sum_{u\in F^{*}} \langle A_{-p(uv)+p(u)/v}f , M_{1/v}f \rangle. \end{align} $$

Now, for $v= \pm -1$ (in fact, for any $v\in F^{*}$ , but this would not lead to a practically useful bound), it is easy to see that

(4.7) $$ \begin{align} \frac{1}{|F^{*}|} \sum_{u\in F^{*}} \langle A_{-p(uv)+p(u)/v}f , M_{1/v}f \rangle \leq \Vert f\Vert_2^2. \end{align} $$

However, for any $v\in F^{*}, v\neq \pm 1$ , we have

(4.8) $$ \begin{align} \bigg| \frac{1}{|F^{*}|} \sum_{u\in F^{*}} \langle A_{-p(uv)+p(u)/v}f , M_{1/v}f \rangle \bigg| \leq \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} A_{-p(uv)+p(u)/v}f\bigg\Vert_2 \Vert f\Vert_2. \end{align} $$

Moreover,

(4.9) $$ \begin{align} &\bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} A_{-p(uv)+p(u)/v}f\bigg\Vert_2\nonumber\\ &\quad \leq \bigg\Vert \frac{|F|}{|F^{*}|}\frac{1}{|F|} \sum_{u\in F} A_{-p(uv)+p(u)/v}f\bigg\Vert_2 + \bigg\Vert \frac{1}{|F^{*}|} A_{-p(0)+p(0)/v}f\bigg\Vert_2. \end{align} $$

However, if $v\not \in \{0,1,-1\}$ , then $-p(uv)+p(u)/v$ is a polynomial of the same degree as $p(u)$ , and so by Proposition 4.2 and because $P_Af=0$ , (4.9) becomes

$$ \begin{align*} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} A_{-p(uv)+p(u)/v}f\bigg\Vert_2 \leq \frac{q}{|F^{*}|^{1/2^{q-1}}}\Vert f\Vert_2. \end{align*} $$

(We used that $|F| / |F^{*}| ( \sqrt {q-1} / |F|^{1/2^{q-1}} )+1 / |F^{*}| \leq q / |F^{*}|^{1/2^{q-1}}$ , whenever $|F|\geq 3$ .) Using this in (4.8), we get that (for $v\notin \{0,1,-1\})$

(4.10) $$ \begin{align} \frac{1}{|F^{*}|} \sum_{u\in F^{*}} \langle A_{-p(uv)+p(u)/v}f , M_{1/v}f \rangle \leq \frac{q}{|F^{*}|^{1/2^{q-1}}}\Vert f\Vert_2^2. \end{align} $$

Combining (4.7) and (4.10), it follows from (4.6) that

$$ \begin{align*} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-p(u)}f\bigg\Vert_2^2 \leq (q+2)\Vert f\Vert_2^2/|F^{*}|^{1/2^{q-1}}. \end{align*} $$

It is shown in [3, proof of Theorem 5.1] that $\Vert f\Vert _2 \leq \sqrt {2\mu (C)}$ . Therefore, the latter inequality readily implies (4.5) and so we conclude.

Theorem 4.4. Let F be a finite field and assume that $\mathcal {A}_F$ acts on $(X,\mathcal {X},\mu )$ by measure-preserving transformations. Let also $p(x) \in F[x]$ be an admissible polynomial of degree $q:=\deg (p(x))$ . Then, for any set $B\in \mathcal {X}$ , such that $(\mu (B))^2> 2(q+2) / |F^{*}|^{1/2^{q-1}}$ , there exists $u\in F^{*}$ so that $\mu (B\cap M_uA_{-p(u)}B)>0.$

If, in addition, the action of $S_A$ is ergodic, then for any sets $B,C \in \mathcal {X}$ which satisfy $\mu (B)\mu (C)> 2(q+2) / |F^{*}|^{1/2^{q-1}}$ , there is some $u\in F^{*}$ with $\mu (B\cap M_uA_{-p(u)}C)>0.$

Remark 4.5. For the case $p(x)=x$ , that is, when $q=1$ , the bounds in this statement coincide with those that Bergelson and Moreira found in [3].

Proof. Let $B,C \in \mathcal {X}$ . For the second conclusion, it suffices to prove the following averages are positive (for the first conclusion, we prove the same thing with $B=C$ ):

(4.11)

where . Now, we observe that

(4.12)

If $S_A$ acts ergodically, then and so (4.12) becomes

(4.13)

If $B=C$ and we do not assume ergodicity, then , where P is the projection onto the space of functions invariant under $\mathcal {A}_F$ by Proposition 4.1. Therefore, $P1=1$ and it follows by the Cauchy–Schwarz inequality that

(4.14)

For the last averages in (4.11), another application of Cauchy–Schwarz’s inequality gives that

(4.15)

So, from (4.5) in Proposition 4.3, the inequality in (4.15) now becomes

In conclusion, (4.11) implies that

(4.16)

As we have alluded to in the beginning of this proof, there are now two routes. If $S_A$ acts ergodically, then (4.16) becomes

(4.17) $$ \begin{align} \frac{1}{|F^{*}|} \sum_{u\in F^{*}} \mu(B\cap M_uA_{-p(u)}C) \geq \mu(B)\mu(C) - \sqrt{2(q+2)\mu(B)\mu(C)} / |F^{*}|^{1/2^{q}}, \end{align} $$

and this is positive whenever $\mu (B)\mu (C)> 2(q+2) / |F^{*}|^{1/2^{q-1}} $ . If $B=C$ and we do not assume ergodicity, then we have

(4.18) $$ \begin{align} \frac{1}{|F^{*}|} \sum_{u\in F^{*}} \mu(B\cap M_uA_{-p(u)}B) \geq (\mu(B))^2 - \sqrt{2(q+2)}\mu(B) / |F^{*}|^{1/2^{q}}, \end{align} $$

which is positive precisely when $(\mu (B))^2> 2(q+2) / |F^{*}|^{1/2^{q-1}}.$

Some quantitative bounds for the set of return times in the previous theorem—which will be used in the proof of Theorem 1.15 given below and in §5—are the following.

Corollary 4.6. Let F be a finite field and assume that $\mathcal {A}_F$ acts on $(X,\mathcal {X},\mu )$ by measure-preserving transformations. Let also $p(x) \in F[x]$ be an admissible polynomial of degree q, $B \in \mathcal {X}$ and $\delta < \mu (B)$ . Then, the set of return times $D:=\{u\in F^{*}: \mu (B \cap M_uA_{-p(u)}B)>\delta \}$ satisfies

(4.19) $$ \begin{align} \frac{|D|}{|F^{*}|} \geq \frac{(\mu(B))^2 - \sqrt{2(q+2)}\mu(B) / |F^{*}|^{1/2^{q}} - \delta}{\mu(B)}. \end{align} $$

If, in addition, the action of $S_A$ is ergodic, then for any $B,C \in \mathcal {X}$ and $\delta < \min {\{\mu (B),\mu (C)\}}$ , the set $D':=\{u\in F^{*}: \mu (B \cap M_uA_{-p(u)}C)>\delta \}$ satisfies

(4.20) $$ \begin{align} \frac{|D'|}{|F^{*}|} \geq \frac{\mu(B)\mu(C) - \sqrt{2(q+2)\mu(B)\mu(C)} / |F^{*}|^{1/2^{q}} - \delta}{\min{\{\mu(B),\mu(C)\}}}. \end{align} $$

Proof. By (4.18), we know that

$$ \begin{align*} \frac{1}{|F^{*}|} \sum_{u\in F^{*}} \mu(B\cap M_uA_{-p(u)}B) \geq (\mu(B))^2 - \sqrt{2(q+2)}\mu(B) / |F^{*}|^{1/2^{q}}. \end{align*} $$

At the same time, $\mu (B \cap M_uA_{-p(u)}B) \leq \mu (B)$ implies that

$$ \begin{align*}\frac{1}{|F^{*}|} \sum_{u\in F^{*}} \mu(B \cap M_uA_{-p(u)}B) \leq \frac{|D|}{|F^{*}|}\mu(B) + \bigg(1-\frac{|D|}{|F^{*}|} \bigg)\delta=\delta + \frac{|D|}{|F^{*}|}(\mu(B)-\delta).\end{align*} $$

Combining the two inequalities, we see that

$$ \begin{align*}(\mu(B))^2 - \sqrt{2(q+2)}\mu(B) / |F^{*}|^{1/2^{q}} \leq \delta + \frac{|D|}{|F^{*}|}(\mu(B)-\delta) \end{align*} $$

and thus

$$ \begin{align*}\frac{|D|}{|F^{*}|}\mu(B) \geq (\mu(B))^2 - \sqrt{2(q+2)}\mu(B) / |F^{*}|^{1/2^{q}} - \delta,\end{align*} $$

which is (4.19). For the ergodic case, we use (4.17) instead of (4.18) and the rest is similar.

We shall conclude this section by proving Theorem 1.15.

Theorem 1.14. Let F be a finite field. Then, if $p(x)\in F[x]$ is an admissible polynomial over F of degree $q:=\deg (p(x))$ and $B_1,B_2 \subset F$ with $|B_1||B_2|> 2(q+2)|F|^{2-(1/2^{q-1})}$ , there are $u,v\in F^{*}$ , so that $vu \in B_1$ and $p(u)+v \in B_2$ .

Remark 4.7. To give a better taste of the bounds, if we are looking for patterns of the form $\{uv , u+v^2\}$ in a subset E of a field of order $3^6=729$ , then our method demands that $|E|>2\sqrt {2 \ 3^{9}} \approx 396$ , and for a field of order $3^7=2187$ , that $|E|>2\sqrt {2} \ 3^{21/4} \approx 904$ .

Proof. Consider the action by affine transformations of $\mathcal {A}_{F}$ on F with the normalized counting measure $\mu $ , that is, $\mu (B)=|B|/|F|$ for any $B\subset F$ . Then, the action of $S_A$ is ergodic. Now, for $s<\min {\{|B_1|,|B_2|\}}$ , we let $\delta =s/|F|$ and $D:=\{u\in F^{*}: \mu (B_1 \cap M_uA_{-p(u)}B_2)>\delta \}.$ By Corollary 4.6, we know that

$$ \begin{align*} \frac{|D|}{|F^{*}|} \geq \frac{\mu(B_1)\mu(B_2) - \sqrt{2(q+2)\mu(B_1)\mu(B_2)} / |F^{*}|^{1/2^{q}} - \delta}{\min{\{\mu(B_1),\mu(B_2)\}}}. \end{align*} $$

This means that

(4.21) $$ \begin{align} |D| \geq \frac{|B_1||B_2||F^{*}|/|F| - |F^{*}|^{1-1/2^q}\sqrt{2(q+2)|B_1||B_2|}- s|F^{*}|}{\min{\{|B_1|,|B_2|\}}}. \end{align} $$

Observe that for $u\in D$ , we have that

$$ \begin{align*}\frac{s}{|F|}=\delta \leq \mu(B_1 \cap M_uA_{-p(u)}B_2) = \frac{| M_{1/u}B_1 \cap A_{-p(u)}B_2 |}{|F|},\end{align*} $$

which means that for each $u\in D$ , there are s elements $v\in F$ , such that $vu\in B_1$ and $v+p(u)\in B_2$ .

5 A ‘colouring trick’ and partition regularity for finite fields

In this section, we will adapt the infinite ‘colouring trick’ presented in [3, §4] to recover a partition regularity result for finite fields, namely Theorem 1.16, from weaker density results established in §4; essentially from the proof of Theorem 1.15. We recall Theorem 1.16 for convenience.

Theorem 1.15. Let $r,q\in \mathbb N$ be fixed. Then, there is $n(r,q) \in \mathbb N$ , so that for a finite field F with $|F|\geq n(r,q)$ and $\mathrm {char}(F)>q$ , and a polynomial $p(x)\in F[x]$ of $\deg (p(x))=q$ , any colouring $F=C_1 \cup \cdots \cup C_r$ contains monochromatic triples of the form $\{u,p(u)+v,uv\}$ .

Proof. Let $r\in \mathbb N$ , $r>1$ , be fixed and let F be any finite field with $|F| \geq n(r,q)$ , for $n(r,q)$ to be determined later. For an r-colouring of such a field, we can permute the colours if necessary and assume that $|C_1| \geq |C_2| \geq \cdots \geq |C_r|$ . Clearly then, ${|C_1| \geq |F|/ r}$ . Next, we pick a number $1\leq r' \leq r$ in the following manner. If ${|C_2| < |F|/r^4}$ , we set $r'=1$ . Else, we have that $|C_2| \geq |F|/r^4$ and $r'\geq 2$ . Then, we either have that $|C_3| \geq |F|/r^8$ , whence $r'\geq 2$ or not and let $r'=2$ . In this fashion, we set

$$ \begin{align*}r':= \max\{1 \leq j \leq r: |C_1| \geq |F|/r\ , \ |C_2| \geq |F|/r^4\ ,\ \ldots \ , \ |C_{j}| \geq |F|/r^{2^{j}} \}.\end{align*} $$

Let $C=C_1 \times \cdots \times C_{r'}$ . We consider the natural measure-preserving action of $\mathcal {A}_{F}$ on $F^{r'}$ (defined coordinate-wise), with the counting measure $\nu $ given by $\nu (E)=|E|/|F^{r'}|$ for any $E \subset F^{r'}$ . For any $\delta =s / |F^{*}| < \nu (C)$ , let

$$ \begin{align*} D=\{u\in F^{*}: \nu(C \cap M_uA_{-p(u)}C)> \delta \}, \end{align*} $$

the size of which we can bound below by Corollary 4.6, which implies that

(5.1) $$ \begin{align} |D| \geq \frac{(\nu(C))^2|F^{*}| - \nu(C)\sqrt{2(q+2)}|F^{*}|^{1-1/2^q} - s}{\nu(C)}. \end{align} $$

Next, we show that

(5.2) $$ \begin{align} |D|> |F|-(|C_1|+\cdots+|C_{r'}|)=|C_{r'+1}|+\cdots+|C_r|. \end{align} $$

Observe that by the definition of $r'$ , it follows that

(5.3) $$ \begin{align} |C_{r'+1}|+\cdots+|C_r| \leq (r-r')|F|/ r^{2^{(r'+1)}} < |F|/ r^{2^{(r'+1)}-1}. \end{align} $$

Combining (5.1) with (5.3), we see that (5.2) follows from

$$ \begin{align*}\nu(C)|F^{*}| -\sqrt{2(q+2)}|F^{*}|^{1-1/2^q} - s / \nu(C)> |F|/ r^{2^{(r'+1)}-1},\end{align*} $$

or equivalently, that

(5.4) $$ \begin{align} \nu(C)> \sqrt{2(q+2)} / |F^{*}|^{1/2^q} + 1/ r^{2^{(r'+1)}-1} + s / (|F^{*}|\nu(C) ) + 1 / (|F^{*}| r^{2^{(r'+1)}-1} ). \end{align} $$

Using the definition of C and $r'$ , it holds that

$$ \begin{align*}\nu(C)=\frac{|C_1|\cdots |C_{r'}|}{|F^{r'}|} \geq \frac{1}{r} \cdot \frac{1}{r^4}\cdot \frac{1}{r^8} \cdots \frac{1}{r^{2^{r'}}}=\frac{1}{r^{(1+4+8+\cdots+2^{r'})} }.\end{align*} $$

Now, one can see that

$$ \begin{align*}\frac{1}{r^{(1+4+\cdots+2^{r'})} }-\frac{1}{r^{2^{(r'+1)}-1}} = \frac{r^{2^{(r'+1)}-1-(2^{r'}+\cdots+2^2+1)}-1}{r^{2^{(r'+1)}-1}} = \frac{r^2-1}{r^{2^{(r'+1)}-1}},\end{align*} $$

when $r'\geq 2$ . (For $r'\geq 2$ , we have that $2^{r'+1}-(2^{r'}+\cdots +2^2 )=4$ .) If $r'=1$ , then the equation becomes $1/ r-1/ r^3 = ( r^2-1 ) / r^3$ . Finally, (5.4) follows from

(5.5) $$ \begin{align} \frac{r^2-1}{r^{2^{(r'+1)}-1}} \geq \sqrt{2(q+2)} / |F^{*}|^{1/2^q} + s / (|F^{*}|\nu(C) ) + 1 / (|F^{*}| r^{2^{(r'+1)}-1} ), \end{align} $$

which holds for $|F|\geq n(r,q)$ , with $n(r,q)$ large enough, since the right-hand side goes to $0$ as $|F| \to \infty $ for $r,q$ fixed. By (5.2), we know that $D \cap (C_1 \cup \cdots \cup C_{r'}) \neq \emptyset $ as

$$ \begin{align*}| D \cap (C_1 \cup \cdots \cup C_{r'}) | \geq |D|-|C_{r'+1}|-\cdots-|C_r|.\end{align*} $$

Thus, there must exist $u\in C_1 \cup \cdots \cup C_{r'}$ , such that $\nu (C \cap M_uA_{-p(u)}C)> s / |F^{*}|$ . Then, if $u \in C_j$ , for $1 \leq j \leq r'$ , by the definition of C and the measure $\nu $ , we will also have that

(5.6) $$ \begin{align} \frac{| C_j/u \cap ( C_j-p(u)) |}{|F|}=\mu(C_j \cap M_uA_{-p(u)} C_j)> \frac{s}{|F^{*}|} > \frac{s}{|F|} \end{align} $$

and hence $C_j/u \cap (C_j-p(u)) \neq \emptyset $ . This implies the existence of $u,v\in F$ with $u\neq 0$ such that $\{u,p(u)+v,uv\} \subset C_j$ . In particular, for each $u\in D \cap (C_1 \cup \cdots \cup C_{r'})$ , there are, by (5.6), at least s monochromatic triples $\{u,p(u)+v,uv\}$ .

Remark 5.1. The observant reader will have noticed that the proof above actually gives that

$$ \begin{align*}| D \cap (C_1 \cup \cdots \cup C_{r'}) | \geq |F^{*}| \bigg( \frac{r^2-1}{r^{2^{(r'+1)}-1}} - \frac{\sqrt{2(q+2)}}{|F^{*}|^{1/2^q}} - \frac{s}{ |F^{*}|\nu(C) } - \frac{1}{|F^{*}| r^{2^{(r'+1)}-1}} \bigg).\end{align*} $$

Therefore, for any finite field with $|F^{*}|\geq n(r,q)$ , we see that

$$ \begin{align*}| D \cap (C_1 \cup \cdots \cup C_{r'}) | \geq c_{r,q} \cdot |F|,\end{align*} $$

where, whenever $n(r,q)$ is large enough,

$$ \begin{align*}c_{r,q}= \frac{r^2-1}{r^{2^{(r'+1)}-1}} - \frac{\sqrt{2(q+2)}}{n(r,q)^{1/2^q}} - \frac{s}{ n(r,q)\cdot \nu(C) } - \frac{1}{n(r,q)\cdot r^{2^{(r'+1)}-1}}>0\end{align*} $$

is a constant that does not depend on $|F|$ . Using the concluding comments of the previous proof, as $s=\delta |F^{*}|$ , we have a total of $c^{\prime }_{r,q} |F|^2$ monochromatic triples of the form $\{u,u+v,uv\}$ , where $c^{\prime }_{r,q}>0$ is a constant that does not depend on $|F|$ .

6 Proof of Theorem 1.18

Throughout this short section, we will assume that K is a countable field and $(F_N)_{N\in \mathbb N}$ is a double Følner sequence in K. We also let $(T_g)_{g\in \mathcal {A}_K}$ denote an action of $\mathcal {A}_K$ on some probability space $(X,\mathcal {X},\mu )$ by measure-preserving transformations. For reference, our main goal is to prove the next result, part of which was initially stated as Theorem 1.18.

Theorem 6.1. Let K, $(F_N)_{N\in \mathbb N}$ , $(X,\mathcal {X},\mu )$ and $(T_g)_{g\in \mathcal {A}_K}$ be as above. Also, we (crucially) further assume that the action of the additive subgroup $S_A=\{A_u: u\in K\}$ is ergodic. Then, given any $B \in \mathcal {X}$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \mu(B \cap A_{-u}B \cap M_{1/u}B) \geq (\mu(B))^3.\end{align*} $$

If, in addition, the action of $S_M$ is ergodic, then for any $B_1,B_2,B_3 \in \mathcal {X}$ , we have that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \mu(B_1 \cap A_{-u}B_2 \cap M_{1/u}B_3) \geq \mu(B_1)\mu(B_2)\mu(B_3).\end{align*} $$

The proof is based on the following (double) ergodic theorem.

Theorem 6.2. Let K, $(F_N)_{N\in \mathbb N}$ , $(X,\mathcal {X},\mu )$ and $(T_g)_{g\in \mathcal {A}_K}$ be as in the beginning of this section. We further assume that the action of the additive subgroup $S_A$ is ergodic. Then, for any $f,g \in L^{\infty }(X,\mu )$ , we have that

$$ \begin{align*} \lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} M_uA_{-u}f \cdot M_{u}g =P_M g \cdot P_Af, \end{align*} $$

where the limit is in $L^2$ .

Proof. Without loss of generality, we assume that f and g are real-valued functions. We begin by decomposing f as $f=P_Af+\tilde {f}$ , where $\tilde {f}=f-P_Af$ . Then,

(6.1) $$ \begin{align} &\frac{1}{|F_N|} \sum_{u\in F_N} M_uA_{-u}f \cdot M_{u}g\nonumber\\ &\quad = \frac{1}{|F_N|} \sum_{u\in F_N} M_uA_{-u}P_Af \cdot M_{u}g + \frac{1}{|F_N|} \sum_{u\in F_N} M_uA_{-u}\tilde{f} \cdot M_{u}g. \end{align} $$

As $P_Af$ is a constant by the ergodicity of $S_A$ , it follows by (the ergodic) Theorem 2.4 that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} M_uA_{-u}P_Af \cdot M_{u}g = P_Mg \cdot P_Af.\end{align*} $$

Hence, the proof will follow from (6.1) if we can show that

$$ \begin{align*}\lim_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} M_uA_{-u}\tilde{f} \cdot M_{u}g = 0.\end{align*} $$

To this end, we let $a_u=M_uA_{-u}\tilde {f} \cdot M_{u}g$ , for $u\in K^{*}$ . By the van der Corput trick (see Lemma 2.6) for $(K^{*},\cdot )$ , it suffices to show that

(6.2) $$ \begin{align} \lim_{M\to \infty} \frac{1}{|F_M|} \sum_{b\in F_M} \limsup_{N\to \infty} \bigg| \frac{1}{|F_N|} \sum_{u\in F_N} \langle a_{ub},a_u \rangle \bigg| =0. \end{align} $$

To this end, we note that for $b\neq 0$ ,

$$ \begin{align*} \langle a_{ub} , a_u \rangle &= \langle M_{ub}A_{-ub}\tilde{f} \cdot M_{ub}g , M_uA_{-u}\tilde{f} \cdot M_{u}g \rangle \\ & = \langle M_{b}A_{-ub}\tilde{f} \cdot M_{b}g , A_{-u}\tilde{f} \cdot g \rangle = \int_X g \cdot M_{b}g \cdot M_bA_{-ub}\tilde{f} \cdot A_{-u}\tilde{f}\, d\mu, \end{align*} $$

where we have used that $M_v$ preserves $\mu $ . Hence, using the equality $M_uA_v=A_{uv}M_u$ (see (2.1)), for all $u,v\in K^{*}$ , we have

$$ \begin{align*} \frac{1}{|F_N|} \sum_{u\in F_N} \langle a_{ub},a_u \rangle = \frac{1}{|F_N|} \sum_{u\in F_N} \int_X g \cdot M_{b}g \cdot A_{-ub^2}M_b\tilde{f} \cdot A_{-u}\tilde{f}\, d\mu \end{align*} $$

and so it suffices to show that

(6.3) $$ \begin{align} \lim_{M\to \infty} \frac{1}{|F_M|} \sum_{b\in F_M} \limsup_{N\to \infty} \bigg| \frac{1}{|F_N|} \sum_{u\in F_N} \int_X g\cdot M_bg \cdot A_{-u}\tilde{f} \cdot A_{-ub^2}M_b\tilde{f} \bigg| =0. \end{align} $$

By Cauchy–Schwarz’s inequality and Lemma 2.10, the convergence in (6.3) follows from

$$ \begin{align*} \lim_{M\to \infty} \frac{1}{|F_M|} \sum_{b\in F_M} \limsup_{N\to \infty} \bigg\Vert \frac{1}{|F_N|} \sum_{u\in F_N} A_{-u}\tilde{f} \cdot A_{-ub^2}M_b\tilde{f}\bigg\Vert^2_2=0. \end{align*} $$

Now, using Proposition 2.8 with $(G,\cdot )=(K,+)$ and $a_u(b) = A_{-u}\tilde {f} \cdot A_{-ub^2}M_b\tilde {f}$ , for any $u,b\in K$ , $b\neq 0$ , we reduce this to showing that

(6.4) $$ \begin{align} \lim_{M\to \infty} \frac{1}{|F_M|} \sum_{b\in F_M} \limsup_{N\to \infty} \frac{1}{|F_N|} \sum_{u\in F_N} \langle a_{u+d}(b),a_u(b) \rangle = 0 \end{align} $$

for any $d\neq 0$ . As before, we see that

$$ \begin{align*} \langle a_{u+d}(b),a_u(b) \rangle = \int_X A_{u(b^2-1)-d}\tilde{f} \cdot A_{-db^2}M_b\tilde{f} \cdot A_{u(b^2-1)}\tilde{f} \cdot M_{b}\tilde{f}\, d\mu.\end{align*} $$

Now, since $A_{u(b^2-1)-d}\tilde {f} \cdot A_{u(b^2-1)}\tilde {f}=A_{u(b^2-1)}( \tilde {f} \cdot A_{-d}\tilde {f} )$ and for $b\notin \{-1,1\}$ , $p(x)=(b^2-1)x$ is a polynomial of degree $1$ in $K[x]$ , we may use the mean ergodic Theorem 2.4 to obtain that the averages in (6.4) become

(6.5) $$ \begin{align} \lim_{M\to \infty} \frac{1}{|F_M|} \sum_{b\in F_M} \int_X P_A( \tilde{f}\cdot A_{-d}\tilde{f} ) \cdot A_{-db^2}M_{b}\tilde{f} \cdot M_{b}\tilde{f}\, d\mu. \end{align} $$

As $S_A$ is ergodic, the projection $ P_A( \tilde {f}\cdot A_{-d}\tilde {f})$ is a constant and so, using (2.1) and the invariance of $\mu $ under $M_v$ once again, (6.5) becomes

(6.6) $$ \begin{align} \lim_{M\to \infty} \frac{1}{|F_M|} \sum_{b\in F_M} P_A( \tilde{f}\cdot A_{-d}\tilde{f} ) \int_X A_{-db}\tilde{f} \cdot \tilde{f}\, d\mu. \end{align} $$

Because $(F_M)_{M\in \mathbb N}$ is a double Følner sequence in K and $d\neq 0$ , it follows by Proposition 2.2 and the mean ergodic theorem that

$$ \begin{align*}\lim_{M\to \infty} \frac{1}{|F_M|} \sum_{b\in F_M} \int_X A_{-db}\tilde{f} \cdot \tilde{f}\, d\mu =\int_X P_A\tilde{f}\cdot \tilde{f}\, d\mu=0,\end{align*} $$

by the definition of $\tilde {f}$ . Therefore, the limit in (6.6) equals zero and so (6.2) follows.

From Theorem 6.2, we can readily recover Theorem 6.1.

Proof of Theorem 6.1

For $B\in \mathcal {X}$ , we see that

as in the proof of Corollary 3.1. By Theorem 6.2 for , this limit becomes

(6.7)

because , $P_M$ is an orthogonal projection and $P_M1=1$ .

For the second part, if in addition $S_M$ acts ergodically, then and the same method gives the result.

7 Generalization of Shkredov’s theorem

This section is devoted to the proof of Theorem 1.20, which generalizes a result due to Shkredov pertaining to finite fields of prime order, as mentioned in §1.2. We actually prove the following slightly more general theorem.

Theorem 7.1. Let F be any finite field. Let also $B_1,B_2,B_3\subset F^{*}$ be any sets satisfying $|B_1||B_2||B_3|>7|F|^{5/2}$ . Then, there exists $u,v\in F^{*}$ such that $v\in B_1, u+v \in B_2$ and $uv\in B_3$ .

We have stated Theorem 7.1 for subsets of $F^{*}$ because working with an indicator function of a set $B\subset F^{*}$ allows us to use inequalities like $\mu (B) \leq P_Mg(x) \leq (|F|/|F^{*}|) \mu (B)$ for all $x\neq 0$ , which simplifies the proof. However, we do not lose generality as our main result, Theorem 1.20, is an immediate corollary of Theorem 7.1.

Proof that Theorem 7.1 implies Theorem 1.20

Let $B_1,B_2,B_3\subset F$ be any sets satisfying $|B_1||B_2||B_3| \geq 8|F|^{5/2}$ and let $B^{\prime }_i=B_i \cap F^{*} \subset F^{*}$ for $i=1,2,3$ . Then,

$$ \begin{align*}|B^{\prime}_1||B^{\prime}_2||B^{\prime}_3| \geq (|B_1|-1)(|B_2|-1)(|B_3|-1)\end{align*} $$

and the right-hand side is strictly larger than

$$ \begin{align*}|B_1||B_2||B_3|-|B_1||B_2|-|B_1||B_3|-|B_2||B_3| \geq |B_1||B_2||B_3|-3|F|^2 \geq 7|F|^{5/2},\end{align*} $$

where the last inequality holds because $3|F|^2 \leq |F|^{5/2}$ for any field of order at least $9$ . Then, the result follows by an application of Theorem 7.1 for the sets $B^{\prime }_1,B^{\prime }_2,B^{\prime }_3$ .

We now proceed to prove Theorem 7.1. This proof is an effort to ‘finitize’ the proof of Theorem 1.18. However, there are some additional technicalities here, because quantities that vanish in the infinite setting are replaced by ‘error’ terms which are bounded (and go to $0$ asymptotically as $|F|$ increases to $\infty $ ).

As in the infinite setting, the proof of Theorem 7.1 relies on a finitistic version of the (double ergodic) Theorem 6.2, which is stated in Proposition 7.3 below. To ease the discussion, we first prove the following estimate that appears in the proof of the latter.

Throughout, F is a finite field and we consider the action by affine transformations of $\mathcal {A}_{F}$ on F with the normalized counting measure $\mu $ . The action of the additive subgroup, $S_A$ , is ergodic, $P_A$ denotes the orthogonal projection onto the space of functions invariant under $S_A$ , and $P_Af=({1}/{|F|}) \sum _{u\in F} A_uf=\int f\ d\mu $ (see also the beginning of §4).

Proposition 7.2. Let F be any finite field and for some $B \subset F^{*}$ . Then,

$$ \begin{align*}\frac{1}{|F^{*}|} \sum_{v\in F^{*}} \bigg\Vert \frac{1}{|F^{*}|}\sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f\bigg\Vert^2_2 \leq \frac{6}{|F|} \Vert f\Vert^4_2.\end{align*} $$

Proof. By Proposition 2.9, we have that for any $v\in F^{*}$ ,

$$ \begin{align*} \bigg\Vert\!\sum_{u\in F} M_{v}A_{-uv}f \cdot A_{- u}f\bigg\Vert^2_2=\sum_{u,w\in F} \langle M_{v}A_{-(u+w)v}f \cdot A_{-(u+w)}f, M_{v}A_{-uv}f \cdot A_{- u}f\rangle. \end{align*} $$

Now, as $M_vA_{-(u+w)v}=A_{-(u+w)v^2}M_v$ and $M_vA_{-uv}=A_{-uv^2}M_v$ , by (2.1) and since $A_{uv^2}$ preserves $\mu $ , we see that

$$ \begin{align*} \bigg\Vert\!\sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f\bigg\Vert^2_2=\sum_{u,w\in F} \langle A_{-wv^2}M_vf \cdot A_{u(v^2-1)-w}f, M_vf \cdot A_{u(v^2-1)}f\rangle. \end{align*} $$

Observe that we can rewrite this as

(7.1) $$ \begin{align} \bigg\Vert\!\sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f\bigg\Vert^2_2 = \sum_{u,w\in F} \langle A_{u(v^2-1)} (f \cdot A_{-w}f), M_v (f \cdot A_{-wv} f) \rangle. \end{align} $$

Whenever $v^2\neq 1$ , we have that

(7.2) $$ \begin{align} &\sum_{u,w\in F} \langle A_{u(v^2-1)} (f \cdot A_{-w}f), M_v (f \cdot A_{-wv} f) \rangle \nonumber\\&\quad = \sum_{w\in F} \langle |F| \cdot P_A(f \cdot A_{-w}f), M_v (f \cdot A_{-wv} f) \rangle \quad \text{by definition of }P_A \nonumber\\&\quad=\sum_{w\in F} |F| \cdot \int_X f \cdot A_{-w}f\, d\mu \int_X M_v (f \cdot A_{-wv} f) \, d\mu \quad \text{by ergodicity of }S_A \nonumber\\&\quad=\sum_{w\in F} |F| \cdot \int_X f \cdot A_{-w}f\ d\mu \int_X f \cdot A_{-wv} f \, d\mu \quad\text{by invariance of }M_v. \end{align} $$

Using (7.2) in (7.1), we see that

(7.3) $$ \begin{align} &\frac{1}{|F^{*}|} \sum_{v\in F^{*}} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f\bigg\Vert^2_2 \nonumber\\&\quad=\frac{|F|}{|F^{*}|^3} \sum_{v\notin \{0,1,-1\}}\sum_{w\in F} \int_X f \cdot A_{-w}f\, d\mu \int_X f \cdot A_{-wv} f \, d\mu \ \nonumber\\&\qquad+\frac{|F|}{|F^{*}|^3}\sum_{w\in F}(\langle f\cdot A_{-w}f, f\cdot A_{-w}f + M_{-1}( f\cdot A_{w}f ) \rangle ). \end{align} $$

Moreover,

(7.4) $$ \begin{align} \sum_{w\in F} \langle f\cdot A_{-w}f, f\cdot A_{-w}f \rangle = \bigg\langle f^2, \sum_{w\in F}A_{-w}f^2 \bigg\rangle = |F| \cdot \Vert f\Vert^4_2 \end{align} $$

and similarly,

(7.5) $$ \begin{align} \sum_{w\in F} \langle f\cdot A_{-w}f, M_{-1}(f\cdot A_{w}f) \rangle = \bigg\langle f\cdot M_{-1}f, \sum_{w\in F} A_{-w} (f\cdot M_{-1}f) \bigg\rangle \leq |F| \cdot \Vert f\Vert^4_2. \end{align} $$

Now, for each $w\neq 0$ , we have that

$$ \begin{align*}\sum_{v\in F} \int_X f \cdot A_{-w}f\, d\mu \int_X f \cdot A_{-wv} f \, d\mu = \int_X f \cdot A_{-w}f\ d\mu \int_X f \cdot P_Af \, d\mu =0 \end{align*} $$

and this immediately gives that

$$ \begin{align*}\sum_{v\in F}\sum_{w\in F} \int_X f \cdot A_{-w}f\, d\mu \int_X f \cdot A_{-wv} f \, d\mu=\sum_{v\in F} \bigg(\int_X f^2\, d\mu\bigg)^2=|F|\cdot \Vert f\Vert^4_2.\end{align*} $$

Therefore, noting that $\sum _{v\notin \{0,1,-1\}}=\sum _{v\in F}-\sum _{v\in \{0,1,-1\}}$ and using the latter, we get that

(7.6) $$ \begin{align} &\sum_{v\notin \{0,1,-1\}}\sum_{w\in F} \int_X f \cdot A_{-w}f\, d\mu \int_X f \cdot A_{-wv} f \, d\mu \nonumber\\&\quad=|F|\cdot \Vert f\Vert^4_2 \ - \sum_{v\in \{0,1,-1\}}\sum_{w\in F} \int_X f \cdot A_{-w}f\ d\mu \int_X f \cdot A_{-wv} f \, d\mu \leq 2\cdot |F|\cdot \Vert f\Vert^4_2. \end{align} $$

The last inequality follows because the rightmost sum vanishes for $v=0$ and is non-negative when $v=1$ . In view of (7.6), the equality in (7.3) is replaced by

$$ \begin{align*} &\frac{1}{|F^{*}|} \sum_{v\in F^{*}} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f\bigg\Vert^2_2 \nonumber\\&\quad\leq 2\frac{ |F|^2}{|F^{*}|^3} \Vert f\Vert^4_2 + \frac{|F|}{|F^{*}|^3}\sum_{w\in F}(\langle f\cdot A_{-w}f, f\cdot A_{-w}f + M_{-1}( f\cdot A_{w}f ) \rangle ), \end{align*} $$

and applying (7.4) and (7.5), we finally see that

$$ \begin{align*}\frac{1}{|F^{*}|} \sum_{v\in F^{*}} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f\bigg\Vert^2_2 \leq 2\frac{ |F|^2}{|F^{*}|^3} \Vert f\Vert^4_2 + 2\frac{|F|^2}{|F^{*}|^3}\Vert f\Vert^4_2 \leq \frac{6}{|F|} \Vert f\Vert^4_2,\end{align*} $$

where the last inequality holds whenever $|F|\geq 8$ .

We now prove Proposition 7.3.

Proposition 7.3. Let F be any finite field and let for some $B \subset F^{*}$ , and for some $C \subset F^{*}$ . Then,

(7.7) $$ \begin{align} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-u}f \cdot M_u g \bigg\Vert_2^2 \leq \frac{7}{\sqrt{|F|}}\mu(B)\mu(C). \end{align} $$

Proof. By Proposition 2.9 and the fact that $M_u$ preserves $\mu $ for all $u\in F^{*}$ , we see that

$$ \begin{align*}\bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-u}f \cdot M_u g \bigg\Vert_2^2 = \frac{1}{|F^{*}|^2} \sum_{u,v\in F^{*}} \langle M_{v}A_{-uv}f \cdot M_vg\ ,\ A_{-u}f \cdot g \rangle.\end{align*} $$

As all functions are real-valued, the above can be rewritten as

$$ \begin{align*} \bigg\langle g, \frac{1}{|F^{*}|} \sum_{v\in F^{*}} M_vg \cdot \bigg( \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_{v}A_{-uv}f \cdot A_{-u}f \bigg) \bigg\rangle.\end{align*} $$

Hence, using the Cauchy–Schwarz inequality, we see that

(7.8) $$ \begin{align} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-u}f \cdot M_u g \bigg\Vert_2^2 \leq \Vert g\Vert_2 \bigg\Vert \frac{1}{|F^{*}|} \sum_{v\in F^{*}} M_vg \cdot \bigg( \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_{v}A_{-uv}f \cdot A_{-u}f \bigg)\bigg\Vert_2. \end{align} $$

By the triangle inequality, the right-hand side in (7.8) is less than or equal to

$$ \begin{align*} &\Vert g\Vert_2 \bigg\Vert \frac{1}{|F^{*}|} \sum_{v\in F^{*}} M_vg \cdot \bigg( \frac{1}{|F^{*}|} \sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f \bigg)\bigg\Vert_2\\&\quad+\Vert g\Vert_2 \bigg\Vert \frac{1}{|F^{*}|^2} \sum_{v\in F^{*}} M_vg \cdot M_vf \cdot f\bigg\Vert_2. \end{align*} $$

For the latter summand, we have that

$$ \begin{align*}\Vert g\Vert_2 \bigg\Vert \frac{1}{|F^{*}|^2} \sum_{v\in F^{*}} M_vg \cdot M_vf \cdot f\bigg\Vert_2=\frac{1}{|F^{*}|}\Vert g\Vert_2\Vert P_M(f\cdot g)\cdot f\Vert_2\leq \frac{|F|}{|F^{*}|^2}\Vert g\Vert^2_2\Vert f\Vert^2_2, \end{align*} $$

as $g(0)=0$ and so $P_M(f\cdot g) \leq (|F|/|F^{*}|) \langle f , g\rangle \leq (|F|/|F^{*}|) \Vert f\Vert \Vert g\Vert $ , by the comments after Theorem 7.1 and the Cauchy–Schwarz inequality. Therefore, (7.8) is replaced by

(7.9) $$ \begin{align} &\bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-u}f \cdot M_u g \bigg\Vert_2^2 \nonumber\\ &\quad\leq \Vert g\Vert_2 \bigg\Vert \frac{1}{|F^{*}|} \sum_{v\in F^{*}} M_vg \cdot \bigg( \frac{1}{|F^{*}|} \sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f \bigg)\bigg\Vert_2 + \frac{|F|}{|F^{*}|^2}\Vert g\Vert^2_2\Vert f\Vert^2_2. \ \end{align} $$

By an application of Cauchy–Schwarz’s inequality for sums of products, we have that

(7.10) $$ \begin{align} &\bigg\Vert \frac{1}{|F^{*}|} \sum_{v\in F^{*}} M_vg \cdot \bigg( \frac{1}{|F^{*}|} \sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f \bigg)\bigg\Vert^2_2 \nonumber \\ &\quad \leq\int_X \frac{1}{|F^{*}|} \sum_{v\in F^{*}} (M_vg)^2 \cdot \frac{1}{|F^{*}|} \sum_{v\in F^{*}} \bigg( \frac{1}{|F^{*}|} \sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f \bigg)^2\, d\mu \nonumber \\ &\quad = \int_X P_Mg \cdot \frac{1}{|F^{*}|} \sum_{v\in F^{*}} \bigg( \frac{1}{|F^{*}|}\sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f\bigg)^2\, d\mu \nonumber\\ &\quad \leq\frac{|F|}{|F^{*}|}\mu(C)\ \cdot \frac{1}{|F^{*}|} \sum_{v\in F^{*}} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f\bigg\Vert^2_2 \ , \end{align} $$

where the equality holds because . By Proposition 7.2, we see that

$$ \begin{align*}\frac{1}{|F^{*}|} \sum_{v\in F^{*}} \bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F} M_{v}A_{-uv}f \cdot A_{-u}f\bigg\Vert^2_2 \leq \frac{6}{|F|} \Vert f\Vert^4_2.\end{align*} $$

Using this in (7.10) and the bound in (7.9), we have that

$$ \begin{align*} &\bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-u}f \cdot M_u g \bigg\Vert_2^2 \leq \frac{\sqrt{6}}{\sqrt{|F|}}\frac{\sqrt{|F|}}{\sqrt{|F^{*}|}}\Vert g\Vert^2_2\Vert f\Vert^2_2\\ &\quad+ \frac{|F|}{|F^{*}|^2}\Vert g\Vert^2_2 \Vert f\Vert^2_2 \leq \frac{\sqrt{6}+1}{\sqrt{|F^{*}|}}\Vert g\Vert^2_2\Vert f\Vert^2_2. \end{align*} $$

Finally, it follows by the definition of f that $\Vert f\Vert ^2_2 \leq 2\mu (B)$ , as shown in [3, proof of Theorem 5.1]. In conclusion, (7.9) becomes

$$ \begin{align*}\bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-u}f \cdot M_u g \bigg\Vert_2^2 \leq \frac{8}{\sqrt{|F|}}\mu(B)\mu(C),\end{align*} $$

since $2(\sqrt {6}+1) \sqrt {|F| / |F^{*}|} \leq 8,$ whenever $|F|\geq 8$ .

We are finally in the position to prove the main result of this section, Theorem 7.1.

Proof of Theorem 7.1

The assumption in the statement of the theorem can be rewritten as $\mu (B_1)\mu (B_2)\mu (B_3)> 7 / \sqrt {|F|}$ and its conclusion is equivalent to the existence of $u\in F^{*}$ so that $\mu (B_1 \cap A_{-u}B_2 \cap M_{1/u}B_3)>0,$ by the definition of $\mu $ . It will thus suffice to show that $\sum _{u \in F^{*}} \mu (B_1 \cap A_{-u}B_2 \cap M_{1/u}B_3)>0.$ Using the fact that $M_u$ preserves $\mu $ for all $u\in F^{*}$ , this is equivalent to

(7.11)

We let . Observe that $P_Af=0$ and is a constant. Then,

(7.12)

As $B_1 \subset F^{*}$ , it follows by the comments after Theorem 7.1 that

Using this in (7.12), we reduce (7.11) to showing that

Applying the Cauchy–Schwarz inequality, the latter follows from showing that

(7.13)

In Proposition 7.3, we showed that

Since , we see that (7.13) holds whenever ${\sqrt {7}}/{|F|^{1/4}}\sqrt {\mu (B_1)\mu (B_2)\mu (B_3)} < \mu (B_1)\mu (B_2)\mu (B_3),$ which is equivalent to our assumption and so we conclude.

As a corollary of the proof, we get the following quantitative result.

Corollary 7.4. Let F be any finite field. Let also $B_1,B_2,B_3\subset F^{*}$ be any sets satisfying $|B_1||B_2||B_3|>7|F|^{5/2}$ . Then, for each $s<\ell :=\min {\{|B_1|,|B_2|,|B_3|\}}$ , there is a set ${D\subset F^{*}}$ of cardinality

$$ \begin{align*} |D| \geq \frac{|B_1||B_2||B_3||F^{*}| / |F|^2-\sqrt{7|B_1||B_2||B_3||F^{*}|^2 / |F|^{3/2}}-s|F^{*}|}{\ell},\end{align*} $$

so that for each $u\in D$ , there are s choices for $v\in F$ such that $v\in B_1$ , $u+v\in B_2$ and $uv\in B_3$ .

Proof. Let $\delta =s/|F|$ for s as above and let

$$ \begin{align*}D=\{u\in F^{*}: \mu(B_3 \cap M_uA_{-u}B_2 \cap M_uB_1)>\delta\}.\end{align*} $$

Similarly to the proof of Corollary 4.6, it follows from the proof of Theorem 7.1 that

(7.14) $$ \begin{align} \frac{|D|}{|F^{*}|} \geq \frac{\mu(B_1)\mu(B_2)\mu(B_3)-\sqrt{7\mu(B_1)\mu(B_2)\mu(B_3) / |F|^{1/2}}-\delta}{m}, \end{align} $$

where $m : = \min {\{ \mu (B_1), \mu (B_2), \mu (B_3) \}}$ . By the definition of $\mu $ , (7.14) is equivalent to

(7.15) $$ \begin{align} |D| \geq \frac{|B_1||B_2||B_3||F^{*}| / |F|^2-\sqrt{7|B_1||B_2||B_3||F^{*}|^2 / |F|^{3/2}}-s|F^{*}|}{\ell}. \end{align} $$

Finally, we see that for each $u\in D$ ,

$$ \begin{align*}&\frac{s}{|F|} \leq \mu(B_3 \cap M_uA_{-u}B_2 \cap M_uB_1)\\ &\quad= \mu(M_{1/u}B_3 \cap A_{-u}B_2 \cap B_1) = \frac{|M_{1/u}B_3 \cap A_{-u}B_2 \cap B_1|}{|F|}\end{align*} $$

and thus, there are s choices for $v \in F$ satisfying $v\in B_1, v+u\in B_2$ and $vu\in B_3$ .

Remark 7.5. The proof of Corollary 7.4 shows in particular that if $A\subset F$ satisfies $|A|\geq \alpha |F|$ for some $\alpha \in (0,1)$ , and $B_1=B_2=B_3=A$ , then $|D|\geq c_{\alpha }|F|$ for some constant $c_{\alpha }>0$ that does not depend on F. This follows by choosing $s=\alpha '|F|$ for some $\alpha ' < \alpha $ and $n\in \mathbb N$ large enough so that the right-hand side in (7.15) is positive whenever $|F|>n$ . Thus, there are $s |D| \geq c^{\prime }_{\alpha }|F|^2$ triples $\{v,v+u,vu\} \subset A$ , where $c^{\prime }_{\alpha }>0$ is another constant that does not depend on $|F|$ .

8 A conditional generalization of Green and Sanders’ theorem

In §5, we devised a finitistic ‘colouring trick’ to prove Theorem 1.16 from Corollary 4.6. Now, using a similar argument and a finitistic version of Conjecture 1.19 as our basis, we will prove a generalization of Green and Sanders’ theorem about ‘monochromatic sums and products’ in finite fields as mentioned in the introduction.

Before stating the aforementioned conjecture, we make another related conjecture that would generalize a special case of Theorem 7.1.

Conjecture 8.1. Let F be any finite field and assume that $\mathcal {A}_{F}$ acts by m.p.t. on a probability space $(X, \mathcal {X}, \nu )$ . Let $B\in \mathcal {X}$ be a set with $\nu (B)> ( c/ |F|)^{a}$ for some constants $a,c>0$ . Then, there exists $u\in F^{*}$ such that

$$ \begin{align*}\nu(B\cap A_{-u}B\cap M_{1/u}B)>0.\end{align*} $$

Remark 8.2. Observe that when $X=F$ and $\nu =\mu $ , the counting measure on F, Theorem 7.1 with $B_1=B_2=B_3$ is a special of this conjecture with $a=1/6$ . However, for this special case, we knew that the additive action of $S_A$ is ergodic, which seems to have been heavily used in the proof of Theorem 7.1, and is no longer true in the general case.

For the purpose of proving the generalization of Green and Sanders’ theorem, that is, Conjecture 1.21, we actually need only consider a special case of Conjecture 8.1 with $X=F^m$ and $\nu =\mu ^m$ , some $m\in \mathbb N$ , where $\mu $ is the counting measure on F, and $B=B_1 \times \cdots \times B_m \subset F^m$ is a set with $\nu (B)> ( c/ |F|)^{a}$ for some constants $a,c>0$ .

A way one could try to prove the aforementioned special case of Conjecture 8.1 would start by decomposing as $P_Ag+f$ , where $f=g-P_Ag$ . Then, following §7 and considering the inner product $\langle f , g\rangle = ({1}/{|F^m|}) \sum _{x\in F^m} f(x)\cdot \overline {g(x)}$ , one would have to show that

(8.1) $$ \begin{align} \frac{1}{|F|}\sum_{u\in F^{*}} \langle g , M_uA_{-u}P_Ag\cdot M_ug \rangle + \frac{1}{|F|}\sum_{u\in F^{*}} \langle g , M_uA_{-u}f\cdot M_ug \rangle> 0. \end{align} $$

This time, $P_Ag$ is not necessarily a constant; however, we still have that

$$ \begin{align*}\frac{1}{|F|}\sum_{u\in F^{*}} \langle g , M_uA_{-u}P_Ag\cdot M_ug \rangle = \langle g , P_M(P_Ag\cdot g) \rangle \geq (\nu(B))^4.\end{align*} $$

Indeed, as $P_Ag \leq 1$ and $P_M$ is an orthogonal projection with $P_M1=1$ , we have

$$ \begin{align*}\langle g , P_M(P_Ag\cdot g) \rangle \geq \langle P_Ag\cdot g , P_M(P_Ag\cdot g) \rangle = \Vert P_M(P_Ag\cdot g)\Vert^2_2 \geq \bigg( \int_{F^m} P_Ag\cdot g\ d\nu \bigg)^2, \end{align*} $$

where the last inequality is Cauchy–Schwarz. Then, arguing similarly for $P_A$ , we have

$$ \begin{align*}\bigg( \int_{F^m} P_Ag\cdot g\, d\nu \bigg)^2 \geq \bigg( \int_{F^m} g\, d\nu \bigg)^4=(\nu(B))^4.\end{align*} $$

Therefore, the proof would follow from the following statement, which is precisely what we are going to use.

Conjecture 8.3. Let F be any finite field and let $m\in \mathbb N$ . Consider the coordinate-wise affine action of $\mathcal {A}_{F}$ by m.p.t. on $(F^m, \nu )$ , where $\nu =\mu ^m=\mu \times \cdots \times \mu $ . Let , where $B=B_1 \times \cdots \times B_m \subset F^m$ and . Then,

$$ \begin{align*}\bigg\Vert \frac{1}{|F^{*}|} \sum_{u\in F^{*}} M_uA_{-u}f \cdot M_ug\bigg\Vert_2 \leq \frac{c}{|F|^b}\Vert f\Vert_2\Vert g\Vert_2\end{align*} $$

for some $b,c>0$ .

As a corollary of Conjecture 8.3, we get the following estimates on the set of return times in the special case of Conjecture 8.1 that we need. The (conditional) proof is a straightforward adjustment of the proof of Corollary 7.4 and so we omit it.

Conjecture 8.4. Let F be a finite field and $m\in \mathbb N$ . Assume that $\mathcal {A}_{F}$ acts on $(F^m,\nu )$ by m.p.t. as above. Let $B=B_1\times \cdots \times B_m \subset F^m$ and $\delta < \nu (B)$ . Then, the set

$$ \begin{align*}D:=\{u\in F^{*}: \nu(B \cap A_{-u}B \cap M_{1/u}B)>\delta\}\end{align*} $$

satisfies

(8.2) $$ \begin{align} \frac{|D|}{|F^{*}|} \geq \frac{(\nu(B))^4 - c\cdot (\nu(B))^{3/2} / |F|^{b} - \delta}{\nu(B)}. \end{align} $$

We are now in a position to apply a version of the finitary ‘colouring trick’ and recover Conjecture 1.21, which we recall for convenience.

Conjecture 1.19. Let $r\in \mathbb N$ be a number of colours. Then, there is $n(r) \in \mathbb N$ , so that for any finite field F with $|F|\geq n(r)$ , any colouring $F=C_1 \cup \cdots \cup C_r$ contains $ d_r|F|^2$ monochromatic quadruples $\{u,v,u+v,uv\}$ , where $d_r>0$ is some constant that does not depend on $|F|$ .

Remark 8.5. Setting $d_r'=d_r/r$ , we get a colour class containing at least $d_r'|F|^2$ monochromatic patterns of the form $\{u,v,u+v,uv\}$ . Moreover, the proof gives an upper bound smaller than $n(r)=r^{4^{(r+2)}}$ for the r-Ramsey number for monochromatic patterns $\{u,v,u+v,uv\}$ in this setting. That is, this conditional proof guarantees that for any r-colouring of a finite field F with $|F| \geq r^{4^{(r+2)}}$ , one of the colours must contain a non-trivial quadruple $\{u,v,u+v,uv\}$ .

Proof. Let $r\in \mathbb N$ , $r>1$ , be fixed and let F be any finite field with $|F| \geq n(r)$ for $n(r)$ to be determined later. For an r-colouring of such a field, we can permute the colours if necessary and assume that $|C_1| \geq |C_2| \geq \cdots \geq |C_r|$ . Clearly, then, $|C_1| \geq |F|/ r$ . Next, we pick a number $1\leq r' \leq r$ in the following manner. If $|C_2| < |F|/r^{16}$ , we set $r'=1$ . Else, we have that $|C_2| \geq |F|/r^{16}$ and $r'\geq 2$ . Then, we either have that $|C_3| \geq |F|/r^{64}$ , whence $r'\geq 2$ or not and let $r'=2$ . Proceeding in this fashion, we set

$$ \begin{align*}r':= \max\{1 \leq j \leq r: |C_1| \geq |F|/r\ , \ |C_2| \geq |F|/r^{16}\ ,\ \ldots \ , \ |C_{j}| \geq |F|/r^{4^{j}} \}.\end{align*} $$

Let $C=C_1 \times \cdots \times C_{r'}$ . We consider the natural measure-preserving action of $\mathcal {A}_{F}$ on $F^{r'}$ (defined coordinate-wise), with the counting measure $\nu $ given by $\nu (E)=|E|/|F^{r'}|$ for any $E \subset F^{r'}$ . So, for $C_1,\ldots ,C_{r'} \subset F$ , we have that $\nu (C_1 \times \cdots \times C_{r'})=\mu (C_1)\cdots \mu (C_{r'})$ , where $\mu $ is the normalized counting measure on F. For any $\delta :=s/ |F^*| < \nu (C)$ , let

$$ \begin{align*} D=\{u\in F^{*}: \nu(C \cap A_{-u}C \cap M_{1/u}C)> \delta \}. \end{align*} $$

Then, by Corollary 8.4, we have that

$$ \begin{align*} \frac{|D|}{|F^{*}|} \geq \frac{(\nu(C))^4 - c\cdot (\nu(C))^{3/2} / |F^{*}|^{b} - \delta}{\nu(C)}, \end{align*} $$

which implies that

(8.3) $$ \begin{align} |D| \geq (\nu(C))^3|F^{*}| - c\cdot |F^{*}|^{1-b} - \frac{|F^{*}|\delta}{\nu(C)}. \end{align} $$

We want to bound below the size of $D \setminus ( C_{r'+1} \cup \cdots \cup C_r )$ , because, for any element u in this set, it holds that $u\in C_1 \cup \cdots \cup C_{r'}$ and also that $\nu (C \cap A_{-u}C \cap M_{1/u}C)> \delta $ . Then, if $u \in C_j$ , for $1 \leq j \leq r'$ , by the definition of C and the measure $\nu $ , we have that $\mu (C_j \cap A_{-u} C_j \cap M_{1/u}C)> \delta $ and hence $|C_j \cap C_j/u \cap (C_j-u)|> s$ , which implies the existence of at least s-elements $v\in F^{*}$ such that $\{u,v,u+v,uv\} \subset C_j$ . To this end, by the choice of $r'$ , we have

(8.4) $$ \begin{align} |C_{r'+1}|+\cdots+|C_r| \leq (r-r')|F|/ r^{4^{(r'+1)}} < |F|/ r^{4^{(r'+1)}-1}. \end{align} $$

Using the definition of C and $r'$ , it holds that

(8.5) $$ \begin{align} \nu(C)=\frac{|C_1|\cdots |C_{r'}|}{|F^{r'}|} \geq \frac{1}{r} \cdot \frac{1}{r^{16}}\cdot \frac{1}{r^{64}} \cdots \frac{1}{r^{4^{r'}}}=\frac{1}{r^{(1+16+64+\cdots+4^{r'})} }. \end{align} $$

Now,

$$ \begin{align*}| D \setminus ( C_{r'+1} \cup \cdots \cup C_r ) | \geq |D|-(|C_{r'+1}|+\cdots+ |C_r| )\end{align*} $$

and so by (8.3), (8.4) and (8.5), we see that

(8.6) $$ \begin{align} | D \setminus ( C_{r'+1} \cup \cdots \cup C_r ) | \geq |F^{*}|/r^{3(1+16+64+\cdots+4^{r'})} - c\cdot |F^{*}|^{1-b} - \frac{|F^{*}|\delta}{\nu(C)}- |F|/ r^{4^{(r'+1)}-1}. \end{align} $$

The quantity at the right-hand side of (8.6) can be rewritten as

$$ \begin{align*}|F^{*}| ( 1/r^{3(1+16+64+\cdots+4^{r'})}-1/ r^{4^{(r'+1)}-1}-\delta/ \nu(C) )- c\cdot |F^{*}|^{1-b}-1/ r^{4^{(r'+1)}-1}.\end{align*} $$

Now, one can see that

$$ \begin{align*}\frac{1}{r^{3(1+16+\cdots+4^{r'})} }-\frac{1}{r^{4^{(r'+1)}-1}} = \frac{r^{4^{(r'+1)}-1-3(4^{r'}+\cdots+4^2+1)}-1}{r^{4^{(r'+1)}-1}} = \frac{r^{12}-1}{r^{4^{(r'+1)}-1}}.\end{align*} $$

(For $r'\geq 2$ , we have that $4^{(r'+1)}-1-3(4^{r'}+\cdots +4^2+1 )=12$ .) Therefore, the right-hand side of (8.6) is greater than or equal to

(8.7) $$ \begin{align} |F^{*}| \bigg( \frac{r^{12}-1}{r^{4^{(r'+1)}-1}}-\delta \cdot r^{(1+16+\cdots+4^{r'})} \bigg)- c\cdot |F^{*}|^{1-b}-1/ r^{4^{(r'+1)}-1}=c_r \cdot |F^{*}|, \end{align} $$

which follows by setting

$$ \begin{align*}c_r = \frac{r^{12}-1}{r^{4^{(r'+1)}-1}}-\delta \cdot r^{(1+16+\cdots+4^{r'})} - c/ |F^{*}|^b -1/ ( |F^{*}|r^{4^{(r'+1)}-1} ).\end{align*} $$

Recall that $|F| \geq n(r)$ . We choose $n(r)$ large enough to guarantee that $c_r>0$ . Since $\delta =s/ |F^{*}|$ and for any $u \in D \setminus ( C_{r'+1} \cup \cdots \cup C_r )$ , we have at least s monochromatic quadruples $\{u,v,u+v,uv\}$ , it follows by (8.7) that there are in total at least

$$ \begin{align*}s \cdot c_r \cdot |F^{*}| =\delta \cdot c_r \cdot |F^{*}|^2 = d_r |F|^2\ \end{align*} $$

monochromatic patterns of the form $\{u,v,u+v,uv\}$ , where $d_r>0$ is a constant that does not depend on the size of F.