Distinct Angles in General Position

TL;DR

This paper improves the upper bounds on the minimum number of distinct angles among points in general position, using geometric properties of a logarithmic spiral to achieve tighter bounds than previous projection-based methods.

Contribution

The authors introduce a novel approach employing logarithmic spiral geometry to significantly tighten bounds on distinct angles in point sets.

Findings

Upper bound for distinct angles reduced to O(n^2)

Bound on subset size with all distinct angles decreased to O(n^{1/2})

Method avoids higher-dimensional projections by using spiral geometry

Abstract

The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat 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. Recent work has introduced bounds on a wide array of variants of this problem, inspired by similar variants in the distance setting. In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by $n$ points in general position from $O(n^{\log_2(7)})$ to $O(n^2)$. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we employ the geometric properties of a logarithmic spiral, sidestepping the need for a projection. We also apply this configuration to reduce the upper bound on the largest integer such that any set of $n$ points in general position has a subset of that size with all distinct angles. This bound is decreased from $O(n^{\log_2(7)/3})$ to $O(n^{1/2})$.