Triangles in the plane and arithmetic progressions in thick compact subsets of upper R Superscript d $\mathbb {R}^d$ R d
Abstract
Abstract This article focuses on the occurrence of 3-point configurations in subsets of upper R Superscript d $\mathbb {R}^d$ R d of sufficient thickness. We prove that a compact set upper A subset of upper R Superscript d $A\subset \mathbb {R}^d$ A ⊂ R d contains a similar copy of any linear 3 $3$ 3 -point configuration (such as a 3 $3$ 3 -point arithmetic progression) provided that A satisfies a mild Yavicoli-thickness condition and an r -uniformity condition for d greater than or equals 2 $d\geq 2$ d ≥ 2 ; or, when d equals 1 $d=1$ d = 1 , the result holds provided that the Newhouse thickness of A is at least 1 $1$ 1 . Moreover, we prove that compact sets upper A subset of upper R squared $A\subset \mathbb {R}^2$ A ⊂
Fields of Study & Subjects
Limits and Structures in Graph Theory
Computational Geometry and Mesh Generation
Analytic Number Theory Research
Publication details
- Journal / source
- Canadian Journal of Mathematics
- Publisher
- Canadian Mathematical Society
- Volume
- 78
- Issue
- 3
- Pages
- 946–986
- Publication date
- 23 January 2026
- Language
- EN