### Abstract

Suppose that d ≥ 2 and m are fixed. For which n is it the case that any n angles can be realised by placing m points in R^{d}? A simple degrees of freedom argument shows that m points in R^{2} cannot realise more than 2m - 4 general angles. We give a construction to show that this bound is sharp when m ≥ 5. In d dimensions the degrees of freedom argument gives an upper bound of dm−(^{d+1}_{2}) general angles. However, the above result does not generalise to this case; surprisingly, the bound of 2m-4 from two dimensions cannot be improved at all. Indeed, our main result is that there are sets of 2m - 3 angles that cannot be realised by m points in any dimension.

Original language | English |
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Pages (from-to) | 995-1012 |

Number of pages | 18 |

Journal | Israel Journal of Mathematics |

Volume | 214 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 1 2016 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Israel Journal of Mathematics*,

*214*(2), 995-1012. https://doi.org/10.1007/s11856-016-1370-1