#30. Equilateral Sets

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Prompts for discussion:

1. What bounds can we get if we are looking for sets of points that are approximately equilateral in two distances? This question combines the setting in this miniature with the one we saw when we discussed the two distances setting. Concretely, a set of points is approximately equal in $p,q$ if the distance between any pair of points lies in $[p - \varepsilon, p + \varepsilon] \cup [q - \varepsilon, q + \varepsilon]$, with $\varepsilon = 1/\sqrt{n}$.

2. Are there other sets with $2d$ points that are equilateral with respect to the $\ell_1$ distance?