The Elekes—Szabó problem and the Uniformity Conjecture

Abstract

In this paper we give a conditional improvement to the Elekes—Szabó problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for F ∈ ℚ[x,y,z] belonging to a particular family of polynomials, and any finite sets A, B, C ⊂ ℚ with ∣A∣ = ∣B∣ = ∣C∣ = n, we have $$\left| {Z\left( F \right) \cap \left( {A \times B \times C} \right)} \right| \ll {n^{2 - {1 \over s}}}.$$ The value of the integer s is dependent on the polynomial F, but is always bounded by s ≤ 5, and so even in the worst applicable case this gives a quantitative improvement on a bound of Raz, Sharir and de Zeeuw [24].We give several applications to problems in discrete geometry and arithmetic combinatorics. For instance, for any set P ⊂ ℚ2 and any two points p1, p2 ∈ ℚ2, we prove that at least one of the pi satisfies the bound $$\left| {\left\{ {\left\| {{p_i} - p} \right\|:p \in P} \right\}} \right| \gg {\left| P \right|^{3/5}},$$ where ∥·∥ denotes Euclidean distance. This gives a conditional improvement to a result of Sharir and Solymosi [28].