Subtended angles

Balister, Paul and Füredi, Zoltán and Bollobás, Béla and Leader, Imre and Walters, Mark (2016) Subtended angles. ISRAEL JOURNAL OF MATHEMATICS, 214 (2). pp. 995-1012. ISSN 0021-2172

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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 Rd? A simple degrees of freedom argument shows that m points in R2 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. © 2016, Hebrew University of Jerusalem.

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