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you'd like. Think of the placing dots evenly spaced around the unit circle (these dots represent the integral roots of unity). You can pack them tighter and tighter (increasing q) and get arbitrarily close to any real valued point on the circle, but of course it will always be an approximation as there will never be a one-to-one map between them.
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This means that for any complex
Hadamard H you can come up with a Butson-type Hadamard G such that |H_{i, j} - G_{i, j}| < \epsilon for as small an \epsilon as you desire, by increasing q. The integral roots of unity are countable so you can't actually make any Hadamard, but you can as close as
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Examples section, what does "in the limit q \to \infty one can approximate all complex Hadamard matrices." is supposed to mean?
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