Improved Elekes-Szabó type estimates using proximity
Jozsef Solymosi, Joshua Zahl
TL;DR
Addressing the real Elekes-Szabó problem for Cartesian products, the paper proves that for an irreducible $f\in\mathbb{C}[x,y,z]$ either $f$ encodes additive group structure or $|(A\times B\times C)\cap Z(f)| \lesssim (|A||B||C|)^{4/7} + |B||C|^{1/2}$ when $|A|\le |B|\le |C|$. It introduces a proximity-based counting technique to control additive quadruples, recasting the problem as point-curve incidences and applying Szemerédi-Trotter-type bounds to obtain the quantitative estimate; an asymmetric variant yields an Elekes-Rónyai type expansion with exponent $3/2$. The results have concrete applications to combinatorial geometry, improving distinct distances bounds in several configurations. The methods bridge $L^2$ energy arguments with incidence geometry via order-driven proximity constraints, offering a new tool for Elekes-Szabó-type problems.
Abstract
We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product $A\times B\times C$ with an algebraic surface $\{f=0\}$ over the reals. In particular, if $A,B,C$ are sets of $N$ real numbers and $f$ is a trivariate polynomial, then either $f$ has a special form that encodes additive group structure (for example $f(x,y,x) = x + y - z$), or $A \times B\times C \cap\{f=0\}$ has cardinality $O(N^{12/7})$. This is an improvement over the previously bound $O(N^{11/6})$. We also prove an asymmetric version of our main result, which yields an Elekes-Ronyai type expanding polynomial estimate with exponent $3/2$. This has applications to questions in combinatorial geometry related to the Erdős distinct distances problem. Like previous approaches to the problem, we rephrase the question as a $L^2$ estimate, which can be analyzed by counting additive quadruples. The latter problem can be recast as an incidence problem involving points and curves in the plane. The new idea in our proof is that we use the order structure of the reals to restrict attention to a smaller collection of proximate additive quadruples.