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Expanding polynomials for sets with additive structure

Sanjana Das, Cosmin Pohoata, Adam Sheffer

Abstract

The expansion of bivariate polynomials is well-understood for sets with a linear-sized product set. In contrast, not much is known for sets with small sumset. In this work, we provide expansion bounds for polynomials of the form $f(x, y) = g(x + p(y)) + h(y)$ for sets with small sumset. In particular, we prove that when $|A|$, $|B|$, $|A + A|$, and $|B + B|$ are not too far apart, for every $\varepsilon > 0$ we have \[|f(A, B)| = Ω\left(\frac{|A|^{256/121 - \varepsilon}|B|^{74/121 - \varepsilon}}{|A + A|^{108/121}|B + B|^{24/121}}\right).\] We show that the above bound and its variants have a variety of applications in additive combinatorics and distinct distances problems. Our proof technique relies on the recent proximity approach of Solymosi and Zahl. In particular, we show how to incorporate the size of a sumset into this approach.

Expanding polynomials for sets with additive structure

Abstract

The expansion of bivariate polynomials is well-understood for sets with a linear-sized product set. In contrast, not much is known for sets with small sumset. In this work, we provide expansion bounds for polynomials of the form for sets with small sumset. In particular, we prove that when , , , and are not too far apart, for every we have We show that the above bound and its variants have a variety of applications in additive combinatorics and distinct distances problems. Our proof technique relies on the recent proximity approach of Solymosi and Zahl. In particular, we show how to incorporate the size of a sumset into this approach.

Paper Structure

This paper contains 33 sections, 27 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our expansion results
  4. Applications to sum-product type bounds
  5. Applications to distinct distances problems
  6. Distinct distances between two lines
  7. Heavy lines in configurations with few distances
  8. Almost tight bounds
  9. Overview of techniques
  10. Preliminaries
  11. Incidence bounds
  12. Irreducibility of polynomials
  13. Cutting curves into monotone pieces
  14. Warmup: a simpler argument without proximity
  15. Outline of the full proof
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $f \in \mathbb R[x, y]$ be a polynomial, and suppose that there do not exist polynomials $p, q \in \mathbb R[x]$ for which Then for all sets $A, B \subseteq \mathbb R$ with $\left\lvert A \right\rvert = \left\lvert B \right\rvert = n$, we have $\left\lvert f(A, B) \right\rvert = \omega(n)$.

Theorems & Definitions (49)

  • Theorem 1.1: Elekes--Rónyai
  • Theorem 1.2: Raz--Sharir--Solymosi
  • Theorem 1.3: Solymosi--Zahl
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Conjecture 1.8: Erdős--Szemerédi
  • Theorem 1.9: Stevens--Warren
  • Corollary 1.10
  • ...and 39 more