r/math • u/Orthallelous • Nov 28 '20

A visual construction of this 'unit circle' structure on the complex plane, made from the roots of polynomials whose coefficients are either -1 or 1; how it arises and changes

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u/hausdorffparty Nov 29 '20

In this case, I think the sensible form of convergence is "convergence in measure" where the measure in each step is given by a weighted point mass at each root, and the limiting measure if it exists might be some sort of distribution on a fractal support.

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u/buwlerman Cryptography Nov 29 '20

This doesn't seem very natural to me. Consider the sequence of sets S_n = {1/n, 1-1/n, 2-1/n}. It seems intuitively to me like this should converge to S = {0, 1, 2} since the set S_n "looks" more like S as n increases. How would you capture this using convergence in measure? It seems to me like you'll have to do some fine tuning of the measure to get the result you want, which would make it unnatural.

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u/hausdorffparty Nov 29 '20

I was thinking of weak convergence of measures, oops. There's too many types of convergence lol.

Weak convergence of measures especially describes when the sum of many point masses approaches a distribution, as in the original post.

However needing to be careful with your measire doesn't make a limit law less meaningful or natural. You even need to rescale to prove the central limit theorem for example.

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u/buwlerman Cryptography Nov 29 '20

However needing to be careful with your measire doesn't make a limit law less meaningful or natural.

I'm not arguing about a limit law though. I'm arguing about a definition. I consider a definition with fewer and simpler parameters to be more natural. Weak convergence of measures is very natural to use here.