Nonstandard methods in combinatorial number theory

    In [MR0472752], Erdős conjectured that if $A\subseteq\mathbb{N}$ has positive lower asymptotic density (in $\mathbb{N}$), then there are infinite subsets $B,C\subseteq\mathbb{N}$ such that $B+C\subseteq A$.

An early partial result is:

Theorem.

(Nathanson [MR0563552]) If $A\subseteq\mathbb{N}$ has positive upper density then, for any $n>0$, there are $B,C\subseteq\mathbb{N}$ such that $B$ is infinite, $C$ has size at least $n$, and $B+C\subseteq A$.

Further progress was made in the setting of countable amenable groups.

Theorem.

(Di Nasso, Goldring, Jin, Leth, Lupini, Mahlburg [MR3361013]) Let $G$ be a countable amenable group.

1. If $A\subseteq G$ has positive Banach density then there are infinite subsets $B,C\subseteq G$ and $h_1,h_2\in G$ such that $B\cdot C\subseteq h_1A\cup h_2A$.
2. If $A\subseteq G$ has Banach density greater than $\frac{1}{2}$ then there are infinite subsets $B,C\subseteq G$ such that $B\cdot C\subseteq A$.

    Let $\mathcal{M}$ be a first-order structure with universe $M$. Given, $m,n>0$, we say a definable subset $R\subseteq M^m\times M^n$ has the independence property if for all $k>0$, there are $(a_i)_{i\in\{1,\ldots,k\}}\subseteq M^m$ and $(b_I)_{I\subseteq\{1,\ldots,k\}}\subseteq M^m$ such that $(a_i,b_I)\in R$ if and only if $i\in I$.

For simplicity, we say “$G$ is an expansion of a group" two mean that $G$ is a first-order structure containing a binary operation satisfying the group axioms. By imposing various notions of “model theoretic tameness" on $G$, we obtain results on the $B+C$ conjecture for definable sets.

Theorem.

(Andrews, Conant, Goldbring [arXiv:1701.07791]) Let $G$ be an expansion of a countable amenable group, and fix a definable set $A\subseteq G$ with positive Banach density.

1. If $G$ is stable then there are infinite $B,C\subseteq G$ such that $B\cdot C\subseteq A$.
2. If $G$ is distal with elimination of $\exists^\infty$ then there are infinite definable $B,C\subseteq G$ such that $B\cdot C\subseteq A$.

Both statements are quick corollaries of known results (see [MR3361013], [arXiv:1612.00908]), and there are many more model theoretic notions of tameness one can consider. However stable and distal are seen as two extreme ends of the class of NIP structures, which we now define.

A structure $\mathcal{M}$ is NIP if, for all $m,n>0$, no definable subset of $M^m\times M^n$ has the independence property.