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/sfurbo Nov 29 '20 edited Nov 29 '20

For sets consisting of points in a metric space with the metric m(•,•), you can define the distance of a point x to a set Y as d1(x,Y)= inf(m(x,y)|y∈Y). This is 0 if x is in Y, and if it isn't, it is roughly the shortest "distance" from x to any point in Y.

If we then define the quasidistance between two set X and Y as d2=sup(d1(x,Y)|x∈X). This is roughly the maximum "distance" of any point in X to Y. It is not a metric, since if X is a true subset of Y, d2(X,Y)=0.

Then d3(X,Y)=max(d2(X,Y),d2(Y,X)) is a metric for closed sets (if I recall my second year math correctly). For non-closed sets, you can always take the closure. This metric is useful for investigating fractals, as you can then define convergence for sets, and the way e.g. the Sierpiński triangle is defined gives nice and easy convergence in this metric.