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Pinned patterns and density theorems in $\mathbb R^d$

Chenjian Wang

TL;DR

The paper addresses the abundance of pinned $k$-point patterns in dense subsets of $\mathbb{R}^d$ and shows a dual picture: a fixed pattern can be avoided at large scales by a carefully constructed dense set, yet a universal positive density phenomenon emerges when one allows a finite catalog of patterns and leverages Szemer\'edi-type structure. The main approach couples a geometric counterexample (thin cone constructions) with a quantitative Szemer\'edi theorem implemented on the torus to force frequency of at least one pattern from a finite catalog across all anchor points. The results advance understanding beyond pairwise distances to multi-point configurations, revealing a combinatorial backbone to geometric pattern occurrence via an isometry-invariant scaling mechanism. The work thereby links geometric configuration problems in Euclidean spaces with deep additive-combinatorial tools, suggesting robust patterns persist under density constraints even in high dimensions.

Abstract

For integers $k\geq 3,d\geq 2,$ we consider the abundance property of pinned $k$-point patterns occurring in $E\subseteq \mathbb R^d$ with positive upper density $δ(E)$. We show that for any fixed $k$-point pattern $V$, there is a set $E$ with positive upper density such that $E$ avoids all sufficiently large affine copies of $V$, with one vertex fixed at any point in $E$. However, we obtain a positive quantitative result, which states that for any fixed $E$ with positive upper density, there exists a $k$-point pattern $V,$ such that for any $x\in E$, the pinned scaling factor set \begin{equation*} D_x^V(E):=\{r> 0: \exists \text{ isometry } O \text{ such that }x+r\cdot O(V)\subseteq E\}, \end{equation*} has upper density $\geq \tilde \varepsilon>0$, where constant $\tilde \varepsilon$ depends on $k,d$ and $δ(E)$.

Pinned patterns and density theorems in $\mathbb R^d$

TL;DR

The paper addresses the abundance of pinned -point patterns in dense subsets of and shows a dual picture: a fixed pattern can be avoided at large scales by a carefully constructed dense set, yet a universal positive density phenomenon emerges when one allows a finite catalog of patterns and leverages Szemer\'edi-type structure. The main approach couples a geometric counterexample (thin cone constructions) with a quantitative Szemer\'edi theorem implemented on the torus to force frequency of at least one pattern from a finite catalog across all anchor points. The results advance understanding beyond pairwise distances to multi-point configurations, revealing a combinatorial backbone to geometric pattern occurrence via an isometry-invariant scaling mechanism. The work thereby links geometric configuration problems in Euclidean spaces with deep additive-combinatorial tools, suggesting robust patterns persist under density constraints even in high dimensions.

Abstract

For integers we consider the abundance property of pinned -point patterns occurring in with positive upper density . We show that for any fixed -point pattern , there is a set with positive upper density such that avoids all sufficiently large affine copies of , with one vertex fixed at any point in . However, we obtain a positive quantitative result, which states that for any fixed with positive upper density, there exists a -point pattern such that for any , the pinned scaling factor set \begin{equation*} D_x^V(E):=\{r> 0: \exists \text{ isometry } O \text{ such that }x+r\cdot O(V)\subseteq E\}, \end{equation*} has upper density , where constant depends on and .
Paper Structure (13 sections, 7 theorems, 87 equations, 3 figures)

This paper contains 13 sections, 7 theorems, 87 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Szemerédi's theorem
  4. Structure of the note
  5. Proof of Theorem \ref{['counterexample']}
  6. Proof of Theorem \ref{['MainTheorem']}
  7. Reduction to "pinned at the origin" version
  8. Adaption of Szemerédi's theorem
  9. Proof of Proposition \ref{['MainLemma']}
  10. Case $d=2$
  11. Case $d\geq 3$
  12. Further directions
  13. Szemerédi's theorem

Key Result

Theorem 1

For $k\geq 3$ and any fixed $k$-point pattern $V=\{p_1,p_2,p_3,...,p_k\}\subseteq \mathbb{R}^d$ that avoids 3 collinear points, there is a set $E$ with positive upper density satisfies that for any $x\in E$, there is $R(x)>0$ such that where

Figures (3)

  • Figure 1: difference between wang_2025_PinnedPatternI and this note
  • Figure 2: Spatial case of $C(\alpha')$
  • Figure 3: Pattern in $xOy$-plane

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Example 1: thin cone
  • Lemma 1
  • proof : Proof of Lemma \ref{['xianrandedongxi']}
  • Proposition 1: main lemma
  • Lemma 2: translation invariance
  • proof : Proof of Theorem \ref{['MainTheorem']} assuming Proposition \ref{['MainLemma']}
  • Lemma 3: size of sets avoiding $(k-1)$-AP
  • ...and 6 more