Abstract
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 \(\Omega (n^{1/5})\) and \(O(n^{(\log _2{7)/3}})\). We also provide bounds on the partite variants of the standard distinct angle problem.
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Notes
We write \(f = O(g)\) if there exists constant C such that \(f(n)\le C g(n)\) for sufficient large n. Conversely, we write \(f=\Omega (g)\) if \(g=O(f)\). Lastly, we write \(f=\Theta (g)\) if \(f=O(g)\) and \(f=\Omega (g)\).
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Acknowledgements
This work was supported by the National Science Foundation Grant 1947438 and Williams College. E. A. Palsson was supported in part by Simons Foundation Grant 360560. The authors would also like to thank their fellow researchers at the SMALL 2021 REU and the anonymous referee for their valuable suggestions.
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Fleischmann, H.L., Hu, H.B., Jackson, F. et al. Distinct Angle Problems and Variants. Discrete Comput Geom 70, 1715–1740 (2023). https://doi.org/10.1007/s00454-022-00466-w
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DOI: https://doi.org/10.1007/s00454-022-00466-w