Improved Elekes-Szab\'o type estimates using proximity

TL;DR

This paper establishes a new bound on the intersection size of Cartesian products with algebraic surfaces, improving previous estimates and applying to combinatorial geometry problems like the Erdős distance problem.

Contribution

It introduces a novel Elekes-Szabó estimate using proximity, achieving a tighter bound and an asymmetric version with applications to expanding polynomials.

Findings

Improved intersection bound from O(N^{11/6}) to O(N^{12/7})

Derived an Elekes-Ronyai type estimate with exponent 3/2

Applied the method to combinatorial geometry problems such as Erdős distances

Abstract

We prove a new Elekes-Szab\'o 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 Erd\H{o}s 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.