Angle chains and pinned variants

TL;DR

This paper investigates bounds on the number of point tuples in a large finite set that form specified angles between successive triples, extending the Erdős unit distance problem to angle configurations and their pinned variants.

Contribution

It introduces new bounds for the maximum number of point configurations with prescribed angles, including both general and pinned cases, advancing understanding of geometric combinatorics.

Findings

Established upper and lower bounds for angle configurations

Analyzed the maximum number of tuples satisfying angle constraints

Extended results to pinned angle variants

Abstract

We study a variant of the Erd\H os unit distance problem, concerning angles between successive triples of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of angles $(\alpha_1,\ldots,\alpha_k)$, we give upper and lower bounds on the maximum possible number of tuples of distinct points $(x_1,\dots, x_{k+2})\in E^{k+2}$ satisfying $\angle (x_j,x_{j+1},x_{j+2})=\alpha_j$ for every $1\le j \le k$ as well as pinned analogues.