Convexity, Squeezing, and the Elekes-Szabó Theorem

TL;DR

The paper investigates how convexity interacts with additive structure by fusing elementary squeezing methods with the Elekes–Szabó incidence bound. It derives sharp expanders combining products and shifts, and shows that, for any finite $A\subset \mathbb{R}$, there exist $a,a'\in A$ with $\left| \dfrac{(aA+1)^{(2)}(a'A+1)^{(2)}}{(aA+1)^{(2)}(a'A+1)} \right| \gtrsim |A|^{31/12}$, while also proving $\max\{|A+A-A|, |A^2+A^2-A^2|, |A^3+A^3-A^3|\} \gtrsim |A|^{19/12}$. Central to the approach is encoding combinatorial growth via non-degenerate polynomials and applying the ES theorem, after verifying non-degeneracy through a derivative test. The paper also demonstrates a two-convex-function enhancement, with $f(x)=x^2$ and $g(x)=x^3$, and discusses possible generalizations to other polynomials, along with the practical challenges of higher-degree cases. The results advance quantitative sum-product-type bounds by integrating elementary squeezing with algebraic incidence geometry, and they suggest promising directions for systematic non-degeneracy criteria and broader function families.

Abstract

This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szabó Theorem in order to give new information. Namely, if we let $A \subset \mathbb R$, we prove that there exist $a,a' \in A$ such that \[\left | \frac{(aA+1)^{(2)}(a'A+1)^{(2)}}{(aA+1)^{(2)}(a'A+1)} \right | \gtrsim |A|^{31/12}.\] We are also able to prove that \[ \max \{|A+A-A|, |A^2+A^2-A^2|, |A^3 + A^3 - A^3|\} \gtrsim |A|^{19/12}.\] Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.

Figures (2)

• Figure 1: This figure provides a pictorial view of the "squeezing" inequalities in \ref{['squeeze1']} and \ref{['squeeze2']}. The blue and red dots are the objects to be counted.
• Figure 2: This figure illustrates the setup for the proof of Theorem \ref{['thm:main22']}. In the proof, we begin by finding many identical intervals, divided into three parts (color-coded in the figure for convenience), with endpoints in $A+A$ which do not overlap. This is illustrated in the top line. Application of our strictly convex function $f$ to these intervals changes their lengths. This is illustrated in the second line. After the application, the lengths of the colored intervals form an increasing sequence as $j$ increases. This puts us in a situation where we can look to apply Lemma \ref{['lem:squeeze2']} for each of these sets of increasing intervals.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1
• proof