Vitaly Bergelson and I have recently uploaded to the arXiv our joint paper `On patterns in large sets of countable fields‘. We prove a result concerning certain monochromatic structures in countable fields and a corresponding density version.

Schur’s Theorem, proved in 1916, states that for any finite coloring of , there exists a monochromatic solution to the equation . In other words, if , there are and such that . It follows from this result that, if , then there are and such that ; this is sometimes called the multiplicative version of Schur’s theorem. (To see how this follows from Schur’s theorem, let for and find on some . This implies that is in ). Moreover, it is known that one can take the same color to contain a configuration and a configuration . However, almost a hundred years later, it is still unknown whether one has a monochromatic configuration for any finite coloring of . Indeed the problem is open even for the configuration .

In this paper we show that for a finite coloring of a countable (infinite) field , there are such that is monochromatic. This result follows from a density statement via a version of the coloring trick I posted about last month. Informally speaking, the density result states that:

Theorem 1Any set with positive density contains a configuration of the form for some .

The tricky part about this statement is what we mean by `positive density’.

It turns out that the regular notion of density doesn’t work. For instance the set of odd numbers in has density , and clearly we can not have for any . The problem is that the configuration we are looking for (namely, ) combines both addition and multiplication. This requires a notion of density which is invariant under addition *and* multiplication. However *no such density exists on *. Indeed the set of even numbers has density with respect to any additively invariant density because . On the other hand, must have density with respect to any multiplicatively invariant density, because .

This is the main reason why our methods don’t work for , but only for countable groups. A more high level reason why there is no density for which is invariant under both addition and multiplication is that the semigroup of all affine transformations of is not amenable. Luckily however, it turns out that the group of affine transformations of a countable field is amenable, and hence there exists a density on invariant under both addition and multiplication. This density is the starting point for our result.

Observe that is equivalent to . We actually prove that if has positive density, then for some .

Another interesting new ingredient in this paper, required for the proof of Theorem 1, is a new version of Furstenberg’s correspondence principle. While previous versions of this principle deal with large subsets of a group, we need to apply it to large subsets of a set where the group is acting. We use this with the affine group acting (canonically) on the field . Thus Theorem 1 will follow from:

Theorem 2Let be a countable field and let be the group of affine transformations of . Suppose is acting by measure preserving transformations on a probability space and has . Then there exists such that

The precise meaning of and requires some notation to define, but intuitively they are the sets that arise from by applying the measure preserving transformation associated with the affine transformation `subtracting ‘ and `dividing by ‘, respectively.

In fact we are able to establish an ergodic theorem for actions of the affine group, which implies that `on average’, the measure is close . As mentioned above, one can feed Theorem 1 (or more precisely, the ergodic theorem just mentioned) to the coloring trick and obtain the coloring result:

Theorem 3Let be a finite partition of a countable field. Then for some there exist (infinitely many) such that .

We are also able to apply the same methods to establish a result similar (but quantitative in nature) to Theorem 1 for finite fields (where the counting measure is invariant under any bijection). This had been done recently for the finite fields (where is a prime) by Shkredov (and actually his result is stronger that what we are able to obtain).

I will now list some open question that we were not able to answer:

- Let . Is there necessarily such that contains a configuration ?
- Let . Is there necessarily such that contains a configuration ?
- Let be a finite partition of a countable field. Is there necessarily such that contains a configuration ?
- Let be a countable field and let be a density invariant under addition and multiplication. If has , must necessarily contain a configuration ?

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