Distinct angles in general position

Abstract

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is the Erdős 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 analogous questions 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(nlog2⁡(7)) to O(n2). We consider a point-set to be in general position if no three points lie on a common line and no four lie on a common circle. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we introduce a construction employing 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(nlog2⁡(7)/3) to O(n1/2).