Elekes-Szabó for groups, and approximate subgroups in weak general position

Elekes-Szabó for groups, and approximate subgroups in weak general position, Discrete Analysis 2023:6, 28 pp.

An important theorem of Elekes and Szabó shows that given an algebraic relation between triples of complex numbers (such as e.g. “being collinear”), if arbitrarily large finite sets of complex numbers can be found with “too many” related triples, then the relation must be of a very specific type: up to some change of coordinates it can be deduced from the graph of the multiplication operation of some one-dimensional algebraic group. More precisely, given an irreducible complex algebraic surface $R$ in $\mathbb C^3$ that is non-degenerate in the sense that its projections to all three pairs of coordinates are dominant, if for every $N\ge 1$ one can find finite sets $A,B,C$ of $N$ complex numbers such that $|A\times B\times C \cap R| = N^{2-o(1)}$ as $N$ tends to infinity, then in fact $R$ is in coordinate-wise finite-to-finite correspondence with the graph $\{(x,y,xy) , x,y \in G\}$ of multiplication of a one-dimensional complex algebraic group $G$.

Elekes and Szabó also generalized their theorem to a multidimensional version, where $A,B$ and $C$ are sets of $N$ points in $\mathbb C^k$ (so the original result was the case $k=1$). In this generalization a crucial assumption of “general position” needs to be made of the sets $A,B,C$. This asserts that the intersection of either $A,B$ or $C$ with a proper algebraic subvariety of $\mathbb C^k$ of bounded degree must be “small”. This smallness can be quantified in various ways leading to slightly different notions of being in general position. The conclusion is again that the algebraic relation is in coordinate-wise finite-to-finite correspondence with the graph of multiplication of some algebraic group $G$, this time of dimension $k$ and not necessarily Abelian (as all one-dimensional groups are).

It was realized in Breuillard and Wang that for a suitable definition of “general position”, the group $G$ in the Elekes-Szabó theorem must in fact be Abelian even in the multidimensional case. If one assumes “strong general position” as in the original paper by Elekes and Szabó, namely that $|A \cap V|$ is bounded as a function of $N$ for every fixed proper algebraic subvariety $V$ of $\mathbb C^k$, or if one only assumes “coarse general position” as was done in a paper of Bays and Breuillard, which requires that $\log |A \cap V| = o(\log N)$ for every fixed proper algebraic subvariety $V$ of $\mathbb C^k$, then it can be shown that $G$ must be abelian.

If on the other hand, one assumes only “weak general position”, namely that $|A \cap V| = O(N^\beta)$ for some $\beta=\beta(V)<1$, then the Elekes-Szabó result no longer holds and the algebraic relation $R$ may no longer “come from” an algebraic group ; a counterexample was given by Bays and Breuillard. However, if we already place ourselves in the situation where $R$ is the graph of multiplication of some complex algebraic group $G$, then the existence of arbitrarily large sets $A$ with “too many related triples” (which essentially means that $A$ is controlled by an “approximate group” $B$ in the sense $|B+B|=O(|B|^{1+o(1)})$) implies that $G$ is nilpotent. This was proved in essence by Breuillard, Green and Tao, and is given another treatment in this paper.

The point of this paper is to establish a converse to the previous statement. That is, given an arbitrary complex nilpotent group, one can construct arbitrarily large approximate subgroups $A$ in weak general position. Interestingly, one cannot do this with the $A$s in coarse general position unless the group $G$ is Abelian. Bays and Breuillard gave such a construction for Abelian groups that even gives coarse general position: the idea was to take generalized arithmetic progressions of increasing length and rank. For nilpotent groups however one cannot let the rank of the progression go to infinity, because the rank of the centre (say in the Heisenberg group) is only marginally smaller than the rank itself. So one has to make do with nilprogressions of large but bounded rank on a generating set made of independent generic elements of G. That this yields a family in weak general position is already not obvious in the Abelian case: for example, when $G$ is a simple Abelian variety, then the fact that large geometric progressions have only a bounded intersection with any given algebraic subvariety is a non-trivial known instance of the Mordell-Lang conjecture. The proof in the paper under review has first to handle the case of general Abelian groups (combinations of vector groups and semiabelian varieties – see Theorem 3.7) and then perform an independent reduction (in Section 4) from the general nilpotent case to the general Abelian case. Along the way the authors have to prove a new result of Mordell-Lang type for general commutative algebraic groups (Propositions 3.3 and 3.7) the proof of which is inspired by Hrushovski’s approach to Mordell-Lang.

This is an interesting contribution to the Elekes-Szabo problem and to the study of nilprogressions in nilpotent algebraic groups, and it also contains a new Mordell-Lang type result for general commutative algebraic groups.