## On {x+y,xy} patterns in large sets of countable fields

Vitaly Bergelson and I have recently uploaded to the arXiv our joint paper On ${\{x+y,xy\}}$ 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 ${{\mathbb N}}$, there exists a monochromatic solution to the equation ${x+y=z}$. In other words, if ${{\mathbb N}=C_1\cup\dots\cup C_r}$, there are ${x,y\in{\mathbb N}}$ and ${i\in\{1,\dots,r\}}$ such that ${\{x,y,x+y\}\subset C_i}$. It follows from this result that, if ${{\mathbb N}=C_1\cup\dots\cup C_r}$, then there are ${x,y\in{\mathbb N}}$ and ${i\in\{1,\dots,r\}}$ such that ${\{x,y,xy\}\subset C_i}$; this is sometimes called the multiplicative version of Schur’s theorem. (To see how this follows from Schur’s theorem, let ${D_j=\{n:2^n\in C_j\}}$ for ${j=1,...,r}$ and find ${\{x,y,x+y\}}$ on some ${D_i}$. This implies that ${\{2^x,2^y,2^x2^y\}}$ is in ${C_i}$). Moreover, it is known that one can take the same color to contain a configuration ${\{x,y,x+y\}}$ and a configuration ${\{x',y',x'y'\}}$. However, almost a hundred years later, it is still unknown whether one has a monochromatic configuration ${\{x,y,x+y,xy\}}$ for any finite coloring of ${{\mathbb N}}$. Indeed the problem is open even for the configuration ${\{x+y,xy\}}$.

In this paper we show that for a finite coloring of a countable (infinite) field ${K}$, there are ${x,y\in K}$ such that ${\{x,x+y,xy\}}$ 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 1 Any set ${E\subset K}$ with positive density contains a configuration of the form ${\{x+y,xy\}}$ for some ${x,y\in K}$.

It turns out that the regular notion of density doesn’t work. For instance the set ${E}$ of odd numbers in ${{\mathbb N}}$ has density ${1/2}$, and clearly we can not have ${\{x+y,xy\}\subset E}$ for any ${x,y\in{\mathbb N}}$. The problem is that the configuration we are looking for (namely, ${\{x+y,xy\}}$) combines both addition and multiplication. This requires a notion of density which is invariant under addition and multiplication. However no such density exists on ${{\mathbb N}}$. Indeed the set ${E}$ of even numbers has density ${1/2}$ with respect to any additively invariant density because ${1=d({\mathbb N})=d(E)+d(E-1)=2d(E)}$. On the other hand, ${E}$ must have density ${1}$ with respect to any multiplicatively invariant density, because ${1=d({\mathbb N})=d(2{\mathbb N})=d(E)}$.

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

Observe that ${\{x+y,xy\}\subset E}$ is equivalent to ${x\in(E-y)\cap(E/y)}$. We actually prove that if ${E}$ has positive density, then ${d\big((E-y)\cap(E/y)\big)>0}$ for some ${y\in K}$.

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 ${K}$. Thus Theorem 1 will follow from:

Theorem 2 Let ${K}$ be a countable field and let ${G}$ be the group of affine transformations of ${K}$. Suppose ${G}$ is acting by measure preserving transformations on a probability space ${(X,\mu)}$ and ${A\subset X}$ has ${\mu(A)>0}$. Then there exists ${y\in K}$ such that ${\mu\big((A-y)\cap(A/y)\big)>0}$

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

In fact we are able to establish an ergodic theorem for actions of the affine group, which implies that `on average’, the measure ${\mu\big((A-y)\cap(A/y)\big)}$ is close ${\mu(A)^2}$. 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 3 Let ${K=C_1\cup\dots\cup C_r}$ be a finite partition of a countable field. Then for some ${i\in\{1,\dots,r\}}$ there exist (infinitely many) ${x,y\in K}$ such that ${\{x,x+y,xy\}\subset C_i}$.

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 ${{\mathbb Z}/p{\mathbb Z}}$ (where ${p}$ 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 ${{\mathbb N}=C_1\cup\dots\cup C_r}$. Is there necessarily ${i\in\{1,\dots,r\}}$ such that ${C_i}$ contains a configuration ${\{x+y,xy\}}$?
• Let ${{\mathbb N}=C_1\cup\dots\cup C_r}$. Is there necessarily ${i\in\{1,\dots,r\}}$ such that ${C_i}$ contains a configuration ${\{x,y,x+y,xy\}}$?
• Let ${K=C_1\cup\dots\cup C_r}$ be a finite partition of a countable field. Is there necessarily ${i\in\{1,\dots,r\}}$ such that ${C_i}$ contains a configuration ${\{x,y,x+y,xy\}}$?
• Let ${K}$ be a countable field and let ${d}$ be a density invariant under addition and multiplication. If ${E\subset K}$ has ${d(E)>0}$, must ${E}$ necessarily contain a configuration ${\{x,x+y,xy\}}$?