Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New discretized polynomial expander and incidence estimates

Ciprian Demeter, William O'Regan

TL;DR

This paper advances discretized incidence geometry by deriving a δ-discretized Elekes–Rónyai-type expander bound for $f(x,y)=x(x+y)$ with an improved exponent $4/3$ in key regimes and by establishing a sharp incidence bound for δ-tubes and δ-squares under non-concentration, governed by the KT and two-ends framework. The main technique translates growth problems into incidence problems between carefully structured sets and employs multi-scale refinements, Frostman discretizations, and two-ends/incidence results to obtain explicit exponents dependent on $s$. The work connects fractal geometry with discrete sum-product phenomena and provides robust tools for analyzing growth in both continuous and discretized settings, with potential generalizations to other semi-diagonal quadratic forms. Overall, it strengthens the bridge between Elekes–Rónyai type expansions and incidence geometry under non-concentration assumptions.

Abstract

We present two applications of recent developments in incidence geometry. One is a $δ$-discretized version of a particular `Elekes--Rónyai' expander problem. For the polynomial we consider, our exponent matches the best known exponent from the discrete case. The second application is an incidence estimate addressing the scenario when both tubes, squares and their shadings satisfy non-concentration assumptions.

New discretized polynomial expander and incidence estimates

TL;DR

This paper advances discretized incidence geometry by deriving a δ-discretized Elekes–Rónyai-type expander bound for with an improved exponent in key regimes and by establishing a sharp incidence bound for δ-tubes and δ-squares under non-concentration, governed by the KT and two-ends framework. The main technique translates growth problems into incidence problems between carefully structured sets and employs multi-scale refinements, Frostman discretizations, and two-ends/incidence results to obtain explicit exponents dependent on . The work connects fractal geometry with discrete sum-product phenomena and provides robust tools for analyzing growth in both continuous and discretized settings, with potential generalizations to other semi-diagonal quadratic forms. Overall, it strengthens the bridge between Elekes–Rónyai type expansions and incidence geometry under non-concentration assumptions.

Abstract

We present two applications of recent developments in incidence geometry. One is a -discretized version of a particular `Elekes--Rónyai' expander problem. For the polynomial we consider, our exponent matches the best known exponent from the discrete case. The second application is an incidence estimate addressing the scenario when both tubes, squares and their shadings satisfy non-concentration assumptions.
Paper Structure (9 sections, 15 theorems, 147 equations)

This paper contains 9 sections, 15 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Expander results
  3. An incidence estimate
  4. Definitions and notation
  5. Auxiliary results
  6. Proofs of expander results
  7. Proof of Proposition \ref{['prop.ele']}
  8. Proof of Theorem \ref{['thm.elecont']}
  9. Proof of Theorem \ref{['huufwf[gmiu9i96=y]new']}

Key Result

Theorem 1.1

Let $0 < s \leq 1.$ There exists $c_0 = c_0(s) > 0$ so that the following holds for all $\epsilon > 0$ small enough. Let $A, B, \subset [1/2,1]$ be $(\delta,s,\delta^{-\epsilon})$-sets. Let $\mathcal{P} \subset A \times B$ be such that $\#\mathcal{P} > \delta^\epsilon\#A\#B.$ Then

Theorems & Definitions (24)

  • Theorem 1.1: Raz--Zahl
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 2.1
  • Theorem 2.2: wangwufurst
  • Definition 2.3
  • Lemma 2.4
  • ...and 14 more