Distinct Angle Problems and Variants

TL;DR

This paper investigates the minimum number of distinct angles among points in the plane, providing new bounds, configurations, and extending the problem to higher dimensions and variants, advancing understanding in discrete geometry.

Contribution

It introduces new bounds and configurations for the distinct angle problem, including asymptotically optimal point sets and analysis of higher-dimensional variants.

Findings

Number of distinct angles in general position is O(n^{log_2(7)}).

Introduces asymptotically optimal point configurations with no four cocircular points.

Bounds on maximal subsets with unique angles are Ω(n^{1/5}) and O(n^{log_2(7)/3}).

Abstract

The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear points in the plane. The standard problem is already well understood. However, it admits many of the same variants as the distinct distance problem, many of which are unstudied. We provide upper and lower bounds on a broad class of distinct angle problems. We show that the number of distinct angles formed by $n$ points in general position is $O(n^{\log_2(7)})$, providing the first non-trivial bound for this quantity. We introduce a new class of asymptotically optimal point configurations with no four cocircular points. Then, we analyze the sensitivity of asymptotically optimal point sets to perturbation, yielding a much broader class of asymptotically optimal configurations. In higher dimensions we show that a variant of Lenz's construction admits fewer distinct angles than the optimal configurations in two dimensions. We also show that the minimum size of a maximal subset of $n$ points in general position admitting only unique angles is $\Omega(n^{1/5})$ and $O(n^{\log_2(7)/3})$. We also provide bounds on the partite variants of the standard distinct angle problem.