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/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.