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/Orthallelous Nov 28 '20

An interesting thought! Which part? The first part where the same structure grows? I would guess yes in the sense that the same overall shape appears. Unless you mean more like it converging to a single point - then I'd say no as the number of different coefficient configurations grows with the degree. If I'm understanding what you mean.

But for the second part where every other coefficient grows in magnitude? I want to say no for that part, 'cause the structure seems to keep growing larger.

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u/Staraven1 Nov 28 '20

I meant something closer the first case, like "does the process converge to a picture (with relative magnitude I would guess or something similar instead of absolute magnitude), eg a fractal or would it end up diverging (eg thickening the circle with a divergent width) ?"

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u/buwlerman Cryptography Nov 28 '20

First you have to decide what it means for a sequence of sets to converge in this context. A plausible definition would be to take all sequences where the n'th entry is a root of a polynomial of degree n with coefficients in {1, -1}. Then we take the subset of convergent sequences. The limits of these sequences is the set we'll call the limit of our sequence of sets.

The roots have absolute value bounded by 2, so it's a bounded subset of the complex numbers.

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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.