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Large cliques in extremal incidence configurations

Tuomas Orponen, Guangzeng Yi

Abstract

Let $P \subset \mathbb{R}^{2}$ be a Katz-Tao $(δ,s)$-set, and let $\mathcal{L}$ be a Katz-Tao $(δ,t)$-set of lines in $\mathbb{R}^{2}$. A recent result of Fu and Ren gives a sharp upper bound for the $δ$-covering number of the set of incidences $\mathcal{I}(P,\mathcal{L}) = \{(p,\ell) \in P \times \mathcal{L} : p \in \ell\}$. In fact, for $s,t \in (0,1]$, $$ |\mathcal{I}(P,\mathcal{L})|_δ \lesssim_ε δ^{-ε-f(s,t)}, \qquad ε> 0,$$ where $f(s,t) = (s^{2} + st + t^{2})/(s + t)$. For $s,t \in (0,1]$, we characterise the near-extremal configurations $P \times \mathcal{L}$ of this inequality: we show that if $|\mathcal{I}(P,\mathcal{L})|_δ \approx δ^{-f(s,t)}$, then $P \times \mathcal{L}$ contains "cliques" $P' \times \mathcal{L}'$ satisfying $|\mathcal{I}(P',\mathcal{L}')|_δ \approx |P'|_δ|\mathcal{L}'|_δ$, $$|P'|_δ \approx δ^{-s^{2}/(s + t)} \quad \text{and} \quad |\mathcal{L}'|_δ \approx δ^{-t^{2}/(s + t)}.$$

Large cliques in extremal incidence configurations

Abstract

Let be a Katz-Tao -set, and let be a Katz-Tao -set of lines in . A recent result of Fu and Ren gives a sharp upper bound for the -covering number of the set of incidences . In fact, for , where . For , we characterise the near-extremal configurations of this inequality: we show that if , then contains "cliques" satisfying ,
Paper Structure (14 sections, 19 theorems, 149 equations, 2 figures)

This paper contains 14 sections, 19 theorems, 149 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related work and further problems
  3. Outline of the paper
  4. Preliminaries
  5. Notations and $(\delta,s)$-sets
  6. Point-line duality and dyadic tubes
  7. Incidence bounds
  8. Uniform sets
  9. Finding non-concentrated subsets
  10. Proof of Theorem \ref{['main']}
  11. Case $s = t$
  12. Case $s < t$
  13. Case $s>t$
  14. Proofs of Corollary \ref{['cor3']} and Proposition \ref{['prop4']}

Key Result

Theorem 1.5

Let $s, t\in (0,1]$ and $K_{P}, K_{\mathcal{L}}\geq 1$. For every $\epsilon>0$, there exists a constant $C=C(\epsilon,K_{P}, K_{\mathcal{L}})$ such that the following holds. Assume $P \subset [0,1]^{2}$ is a Katz-Tao $(\delta,s, K_{P})$-set and $\mathcal{L} \subset \mathcal{A}(2)$ is a Katz-Tao $(\d where $f(s,t)=\frac{s^2+st+t^2}{s+t}$. Moreover, this bound is sharp up to $C\delta^{-\epsilon}$.

Figures (2)

  • Figure 1: A pair $P \cap \mathcal{L}$ admitting a decomposition into large $\delta$-sub-cliques.
  • Figure 2: The function $f$, and the function determined by the slopes $\sigma_{j}$.

Theorems & Definitions (46)

  • Definition 1.1: $(\delta,\theta)$-clique
  • Definition 1.2: Katz-Tao $(\delta,s,C)$-set
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.10
  • Remark 1.13
  • Remark 1.14
  • Corollary 1.15
  • ...and 36 more