Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2506.00571 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918373409947648 |
|---|---|
| author | Sandberg-Clark, Samantha Taylor, Krystal |
| author_facet | Sandberg-Clark, Samantha Taylor, Krystal |
| contents | This article focuses on the occurrence of 3-point configurations in subsets of $\mathbb{R}^d$ of sufficient thickness. We prove that a compact set $A\subset \mathbb{R}^d$ contains a similar copy of any linear $3$-point configuration (such as a $3$-point arithmetic progression) provided $A$ satisfies a mild Yavicoli-thickness condition and an $r$-uniformity condition for $d\geq 2$; or, when $d=1$, the result holds provided the Newhouse thickness of $A$ is at least $1$.
Moreover, we prove that compact sets $A\subset \mathbb{R}^2$ contain the vertices of an equilateral triangle (and more generally, the vertices of a similar copy of any given triangle) provided $A$ satisfies a mild Yavicoli-thickness condition and an $r$-uniformity condition. Further, $C\times C$ contains the vertices of an equilateral triangle (and more generally the vertices of a similar copy of any given 3-point configuration) provided the Newhouse thickness of $C$ is at least $1$. These are among the first results in the literature to give explicit criteria for the occurrence of 3-point configurations in the plane.These are among the first results in the literature to give explicit criteria for the occurrence of three-point configurations in the plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00571 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Triangles in the Plane and arithmetic progressions in thick compact subsets of $\mathbb{R}^d$ Sandberg-Clark, Samantha Taylor, Krystal Classical Analysis and ODEs Combinatorics This article focuses on the occurrence of 3-point configurations in subsets of $\mathbb{R}^d$ of sufficient thickness. We prove that a compact set $A\subset \mathbb{R}^d$ contains a similar copy of any linear $3$-point configuration (such as a $3$-point arithmetic progression) provided $A$ satisfies a mild Yavicoli-thickness condition and an $r$-uniformity condition for $d\geq 2$; or, when $d=1$, the result holds provided the Newhouse thickness of $A$ is at least $1$. Moreover, we prove that compact sets $A\subset \mathbb{R}^2$ contain the vertices of an equilateral triangle (and more generally, the vertices of a similar copy of any given triangle) provided $A$ satisfies a mild Yavicoli-thickness condition and an $r$-uniformity condition. Further, $C\times C$ contains the vertices of an equilateral triangle (and more generally the vertices of a similar copy of any given 3-point configuration) provided the Newhouse thickness of $C$ is at least $1$. These are among the first results in the literature to give explicit criteria for the occurrence of 3-point configurations in the plane.These are among the first results in the literature to give explicit criteria for the occurrence of three-point configurations in the plane. |
| title | Triangles in the Plane and arithmetic progressions in thick compact subsets of $\mathbb{R}^d$ |
| topic | Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2506.00571 |