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 $\mathbb C^k$, t