## Measure preserving actions of affine semigroups and {x+y,xy} patterns

Vitaly Bergelson and I have recently submitted to the arXiv our paper entitled Measure preserving actions of affine semigroups and ${\{x+y,xy\}}$ patterns’. The main purpose of this paper is to extend the results of our previous paper, establishing some partial progress towards the following (still open) conjecture:

Conjecture 1 Let ${{\mathbb N}=C_1\cup\cdots\cup C_r}$ be an arbitrary finite partition of the natural numbers. Then there exists ${i\in\{1,\dots,r\}}$ and infinitely many ${x,y\in{\mathbb N}}$ such that

$\displaystyle \{x+y,xy\}\subset C_i$

This conjecture is the simplest unknown case of a much stronger conjecture pertaining various monochromatic polynomial configurations, which is itself only a piece in the wide open problem of classifying which polynomial configurations are partition regular over ${{\mathbb N}}$; see this post for another recent result in this direction.

In our previous paper we showed how a relaxation of Conjecture 1 could be solved via ergodic theoretical methods. In particular, we showed that for any finite partition of a ${{\mathbb Q}}$, one of the cells of the partition contain many triples ${\{x+y,xy\}}$ with ${x,y\in{\mathbb Q}}$. Observe that this last result is strictly weaker than Conjecture 1, which is trivially equivalent to the statement that for any finite partition ${{\mathbb Q}=C_1\cup\cdots\cup C_r}$ of the rational numbers, there exists a cell ${i\in\{1,\dots,r\}}$ and infinitely many ${x,y\in{\mathbb N}}$ such that ${\{x+y,xy\}\subset C_i}$. The main (application of the main) result in the new paper is the following intermediate’ result:

Theorem 2 Let ${{\mathbb Q}=C_1\cup\cdots\cup C_r}$ be an arbitrary finite partition of ${{\mathbb Q}}$. Then there exists ${i\in\{1,\dots,r\}}$ and infinitely many ${x\in{\mathbb Q}}$, ${y\in{\mathbb N}}$ such that

$\displaystyle \{x+y,xy\}\subset C_i$

Our methods extend to a significantly more general setup.

Definition 3 A ring ${R}$ is called a Large Ideal Domain (LID) if it is an integral domain and any non-zero ideal has finite index (as an additive subgroup).

Fields are trivially LID, since every (non-zero) ideal is the whole field. It is also easy to see that ${{\mathbb Z}}$ is an LID. More generally the ring of integers of an algebraic number field (for example, the ring of Gaussian integers) is an LID. Yet another example of an LID ring is the ring of polynomials over a finite field.

Theorem 2 follows from the following. (Strictly speaking, Theorem 4 only guarantees that, in the formulation of Theorem 2, one can take ${y\in{\mathbb Z}}$ instead of in ${{\mathbb N}}$. To obtain Theorem 2 as stated we need an additional trick.)

Theorem 4 Let ${K}$ be a countable field, let ${R\subset K}$ be an LID ring and let ${K=C_1\cup\cdots\cup C_r}$ be an arbitrary finite partition of ${K}$. Then there exists ${i\in\{1,\dots,r\}}$ and infinitely many ${x\in K}$, ${y\in R}$ such that

$\displaystyle \{x+y,xy\}\subset C_i$

— 1. Amenability and ultrafilters —

The main reason why working with ${{\mathbb Q}}$ is any easier than with ${{\mathbb N}}$ is the fact that the semigroup of all affine transformations ${{\mathcal A}_{\mathbb N}:=\{x\mapsto ax+b:a,b\in{\mathbb N}\}}$ is not amenable, while the corresponding affine group ${{\mathcal A}_{\mathbb Q}:=\{x\mapsto ax+b:a,b\in{\mathbb Q}; a\neq0\}}$ is. This fact allows one to study measure preserving actions of ${{\mathcal A}_{\mathbb Q}}$ using classical methods (namely Cesàro averages, built upon Følner sequences, which are only available in (some) amenable semigroups).

However, in the past few decades a new method to study the limiting behaviour of measure preserving systems appeared, in the form of IP-convergence and limits along idempotent ultrafilters (discussed, for instance, in this previous post). Remarkably, these methods can be applied in any semigroup, regardless of amenability, and hence provide a possible method to study measure preserving actions of, say, the affine semigroup ${{\mathcal A}_{\mathbb N}}$ of ${{\mathbb N}}$. In this paper we did precisely this and obtained Theorem 5 below regarding measure preserving actions of ${{\mathcal A}_{\mathbb N}}$.

In order to state Theorem 5, I need first to introduce some terminology. For each ${n\in{\mathbb Z}}$, denote by ${A_n\in{\mathcal A}_{\mathbb Z}}$ the map ${x\mapsto x+n}$, and denote by ${M_n\in{\mathcal A}_{\mathbb Z}}$ the map ${x\mapsto xn}$. Denote by ${\beta{\mathbb Z}}$ the Stone-Čech compactification of ${{\mathbb Z}}$, which can be identified with the set of all ultrafilters on ${{\mathbb Z}}$. Recall (for instance, from this previous post) the notion of minimal idempotents in ${\beta{\mathbb Z}}$. A measure preserving action of a semigroup ${G}$ is an action ${(T_g)_{g\in G}}$ on a probability space ${(X,{\mathcal B},\mu)}$ by measurable maps satisfying ${\mu(T_g^{-1}B)=\mu(B)}$ for every ${B\in{\mathcal B}}$ and ${g\in G}$. Given such an action, the compact projection ${V:L^2(X)\rightarrow L^2(X)}$ is the orthogonal projection onto the closed subspace of vectors ${f\in L^2(X)}$ whose orbit ${\{f\circ T_g:g\in G\}}$ is pre-compact. Observe that a measure preserving action of ${{\mathcal A}_{\mathbb Z}}$ induces measure preserving actions of ${({\mathbb Z},+)}$ and ${({\mathbb Z}^*,\times)}$, through the identifications ${n\mapsto A_n}$ and ${n\mapsto M_n}$.

Theorem 5 Let ${p\in\beta{\mathbb Z}}$ be an ultrafilter which is multiplicatively minimal idempotent and belongs to the closure of the additively minimal idempotent ultrafilters. Let ${(T_g)_{g\in{\mathcal A}_{\mathbb Z}}}$ be a measure preserving action of the affine semigroup ${{\mathcal A}_{\mathbb Z}}$ on a probability space ${(X,{\mathcal B},\mu)}$. Then, for every ${f_1,f_2\in L^2(X)}$ we have

$\displaystyle p\,\text{-}\lim_u\big\langle f_1\circ T_{M_n},f_2\circ T_{A_n}\big\rangle=\langle V_Af_1,V_Mf_2\rangle$

where ${V_A}$ and ${V_M}$ are the compact projections of ${({\mathbb Z},+)}$ and ${({\mathbb Z}^*,\times)}$, respectively.

Unfortunately, in our previous paper, the amenability of ${{\mathcal A}_{\mathbb Q}}$ was used not only to study measure preserving actions of the affine group ${{\mathcal A}_{\mathbb Q}}$ but also to reduce the combinatorial problem to a statement about measure preserving actions in the first place! Indeed, the fact that ${{\mathcal A}_{\mathbb Q}}$ is amenable allows one to create affinely invariant means in ${{\mathbb Q}}$ (i.e., linear maps ${\lambda:{\mathcal P}(K)\rightarrow[0,1]}$ which are invariant under both additive shifts and multiplicative shifts). Thanks to (a version of) Furstenberg’s correspondence principle, one can then study any affinely defined configurations (such as ${\{x+y,xy\}}$) in subsets of ${{\mathbb Q}}$ with positive affinely invariant mean by studying measure preserving actions of ${{\mathcal A}_K}$.

On the other had, no affinely invariant mean exists on ${{\mathbb N}}$ (or, in fact, on any integrable domain which is not a field). While limits along ultrafilters have proved good substitutes for Cesàro averages when studying measure preserving systems with non-amenable semigroups, there is no good way to replace invariant means. Therefore, in order to obtain combinatorial applications of Theorem 5 we must embed ${{\mathbb N}}$ (or ${{\mathbb Z}}$) inside ${{\mathbb Q}}$, use the amenability of ${{\mathcal A}_{\mathbb Q}}$ (and, of course, Furstenberg’s correspondence principle) to create a relevant measure preserving action of ${{\mathcal A}_{\mathbb Q}}$ and then restrict our attention to the induced measure preserving action of ${{\mathcal A}_{\mathbb Z}}$. As an outcome of this somewhat contorted way of applying Theorem 5 we obtain Theorem 4 (instead of, more desirably, a solution to Conjecture 1).