A Bilinear Bogolyubov Argument in Abelian Groups

A bilinear Bogolyubov argument in abelian groups, Discrete Analysis 2024:20, 41 pp.

Bogolyubov’s argument states that if $A$ is a dense subset of $\mathbb Z_n$, then $2A-2A=\{x+y-z-w:x,y,z,w\in A\}$ contains a large and highly structured subset known as a Bohr set. If $r_1,\dots,r_k$ are residue classes mod $n$ and $\delta>0$, then the Bohr set $B(r_1,\dots,r_k;\delta)$ is defined to be the set of all $x\in\mathbb Z_n$ such that $|1-\exp(2\pi ir_jx/n)|\leq\delta$ for all $j=1,2,\dots,k$. (There are equivalent definitions, but this one is convenient for the discussion here.) Such sets, when $k$ is small and $\delta$ is not too small (which we get from Bogolyubov’s argument), resemble low-codimensional subspaces of a vector space, and this additive structure has been exploited in many proofs in additive combinatorics.

Bogolyubov’s argument can be straightforwardly generalized to arbitrary finite Abelian groups. If $G$ is a finite Abelian group, $\chi_1,\dots,\chi_k$ are characters on $G$ and $\delta>0$, then the Bohr set $B(\chi_1,\dots,\chi_k;\delta)$ is the set of all $x\in G$ such that $|1-\chi_j(x)|\leq\delta$ for all $j=1,2,\dots,k$. When $G=\mathbb F_p^n$, where the Bohr sets actually are low-codimensional subspaces. That is, if $A\subset\mathbb F_p^n$ has density $\alpha$, then $2A-2A$ contains a subspace of $\mathbb F_p^n$ of codimension at most $C(\alpha)$. The question of how large $C(\alpha)$ needs to be is very interesting: thanks to work of Tom Sanders it is known to be polylogarithmic in $1/\alpha$, but it is conjectured to be logarithmic.

A different kind of generalization of Bogolyubov’s method was obtained by Bienvenu and Lê, and independently by Gowers and the author of this paper, again in the case $G=\mathbb F_p^n$. Speaking very loosely, Bogolyubov’s argument tells us that if we take iterated sumsets of a dense subset of $\mathbb F_p^n$, then the resulting sets will become more and more subspace-like. Suppose now that we take a dense subset $A$ of $\mathbb F_p^n\times\mathbb F_p^n$. Call sections of $A$ of the form $A_{x\bullet}=\{y\in\mathbb F_p^n:(x,y)\in A\}$ the rows of $A$ and sections of the form $A_{\bullet y}=\{x\in\mathbb F_p^n:(x,y)\in A\}$ the columns of $A$. Define the horizontal difference set of $A$ to be the result of replacing each row $A_{x\bullet}$ of $A$ by its difference set $A_{x\bullet}-A_{x\bullet}$, and define the vertical difference set in a similar way for the columns.

Bogolyubov’s argument tells us that if we take the horizontal difference set twice, then all dense rows of $A$ will contain low-codimensional subspaces, and if we take the vertical difference set twice, then all dense columns will. Thus, one might expect that if one takes a suitable composition of these two operations, one will end up with a set that is subspace-like in both the vertical and horizontal directions, though this is not trivial because one needs to show that the operations in one direction do not undo the progress made in the other direction.

What might such a set look like? An instructive example is the following. Let $M$ be a linear map from $\mathbb F_p^n$ to $\mathbb F_p^n$ and let $A$ be the set of all $(x,y)\in\mathbb F_p^n\times\mathbb F_p^n$ such that $x.My=0$, (where $a.b$ stands for the mod-$p$ inner product $\sum_{i=1}^na_ib_i$). Then for each $x$, the row $A_{x\bullet}$ is the set of all $y$ such that $x.My=0$, which is a subspace of codimension 1, except when $M^*x=0$ when it is all of $\mathbb F_p^n$. A similar statement holds for columns, so all the rows and columns of $A$ are subspaces.

Sets of the kind just defined are 1-dimensional bilinear varieties in $\mathbb F_p^n\times\mathbb F_p^n$. Taking an intersection of $k$ of them, we obtain a $k$-dimensional bilinear variety. The theorem of Bienvenu and Lê and of Gowers and Milićević is a bilinear version of Bogolyubov’s argument, which states that if $A$ is a dense subset of $\mathbb F_p^n\times\mathbb F_p^n$ then there is a composition of a bounded number of operations of taking horizontal and vertical difference sets (the bound here is absolute – it does not depend on the density of $A$) such that the resulting set contains a bilinear variety of bounded codimension (here the bound does depend on the density of $A$).

Improved bounds for the bilinear Bogolyubov theorem were later obtained by Hosseini and Lovett.

This paper generalizes the bilinear Bogolyubov argument to arbitrary finite Abelian groups. For some of the proof, it follows the approach of Hosseini and Lovett, but it diverges from it at a point where their argument involves considering ranks of linear combinations of linear maps. Even before the point of divergence, the generalization is not trivial: there are many concepts that are straightforward for $\mathbb F_p^n$ but that need to be generalized rather carefully in order to obtain a proof for arbitrary Abelian groups. It is likely that these definitions and associated tools will be useful for other purposes, such as proving inverse theorems for $U^k$ norms in general Abelian groups.