Distinct Angle Problems and Variants

• Published in: Discrete & computational geometry Vol. 70; no. 4; pp. 1715 - 1740
• Main Authors: Fleischmann, Henry L., Hu, Hongyi B., Jackson, Faye, Miller, Steven J., Palsson, Eyvindur A., Pesikoff, Ethan, Wolf, Charles
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2023; Springer Nature B.V
• Subjects: Angles (geometry); Asymptotic properties; Combinatorics; Computational Mathematics and Numerical Analysis; Configurations; Geometry; Lower bounds; Mathematics; Mathematics and Statistics; Erdős problems; Discrete geometry; 52C30; Maximal subsets; 52C10; 52B11; Angles; 52C35; Restricted point configurations; 52B15
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-022-00466-w

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Less well known is Erdős’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 Ω ( n 1 / 5 ) and O ( n ( log 2 7 ) / 3 ) . We also provide bounds on the partite variants of the standard distinct angle problem.