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}.

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 {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\}}?

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About Joel Moreira

PhD Student at OSU in Mathematics. I'm portuguese.
This entry was posted in Combinatorics, Ergodic Theory, paper, Ramsey Theory and tagged , , , , , , , , , , . Bookmark the permalink.

3 Responses to On {x+y,xy} patterns in large sets of countable fields

  1. Pingback: Additive vs multiplicative densities | I Can't Believe It's Not Random!

  2. Pingback: Description of my work using few words | I Can't Believe It's Not Random!

  3. Pingback: Measure preserving actions of affine semigroups and {x+y,xy} patterns | I Can't Believe It's Not Random!

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