Vitaly Bergelson and I have recently submitted to the arXiv our paper entitled `Measure preserving actions of affine semigroups and 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 1Let be an arbitrary finite partition of the natural numbers. Then there exists and infinitely many such that

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 ; 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 , one of the cells of the partition contain many triples with . Observe that this last result is strictly weaker than Conjecture 1, which is trivially equivalent to the statement that for any finite partition of the rational numbers, there exists a cell and infinitely many such that . The main (application of the main) result in the new paper is the following `intermediate’ result:

Theorem 2Let be an arbitrary finite partition of . Then there exists and infinitely many , such that

Our methods extend to a significantly more general setup.

Definition 3A ring is called aLarge 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 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 instead of in . To obtain Theorem 2 as stated we need an additional trick.)

Theorem 4Let be a countable field, let be an LID ring and let be an arbitrary finite partition of . Then there exists and infinitely many , such that

** — 1. Amenability and ultrafilters — **

The main reason why working with is any easier than with is the fact that the semigroup of all affine transformations is not amenable, while the corresponding affine group is. This fact allows one to study measure preserving actions of 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 of . In this paper we did precisely this and obtained Theorem 5 below regarding measure preserving actions of .

In order to state Theorem 5, I need first to introduce some terminology. For each , denote by the map , and denote by the map . Denote by the Stone-Čech compactification of , which can be identified with the set of all ultrafilters on . Recall (for instance, from this previous post) the notion of minimal idempotents in . A *measure preserving action* of a semigroup is an action on a probability space by measurable maps satisfying for every and . Given such an action, the *compact projection* is the orthogonal projection onto the closed subspace of vectors whose orbit is pre-compact. Observe that a measure preserving action of induces measure preserving actions of and , through the identifications and .

Theorem 5Let be an ultrafilter which is multiplicatively minimal idempotent and belongs to the closure of the additively minimal idempotent ultrafilters. Let be a measure preserving action of the affine semigroup on a probability space . Then, for every we havewhere and are the compact projections of and , respectively.

Unfortunately, in our previous paper, the amenability of was used not only to study measure preserving actions of the affine group but also to reduce the combinatorial problem to a statement about measure preserving actions in the first place! Indeed, the fact that is amenable allows one to create affinely invariant means in (i.e., linear maps 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 ) in subsets of with positive affinely invariant mean by studying measure preserving actions of .

On the other had, no affinely invariant mean exists on (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 (or ) inside , use the amenability of (and, of course, Furstenberg’s correspondence principle) to create a relevant measure preserving action of and then restrict our attention to the induced measure preserving action of . 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).