Let's say we have an infinite nested radical with random terms (positive integers) which can take on finitely many values in the range $(n_1,n_2)$. What would the distribution look like for these radicals if all terms appear with equal probability?
I naively expected it to look, well, random, even for $n_2=n_1+1$. However, this is not the case.
See how the distribution looks for infinite radicals with random terms which can be either $1$ or $2$ (number of terms is $300$ in each radical, size of the distribution is $700$, the histogram contains $250$ bins, and I used Mathematica RandomInteger):
It's obvious, that the numbers can't be smaller than the golden ratio:
And can't be larger than $2$:
There are also two more 'attractors' for this distribution:
What I can't explain is the occurence of 'doublets' for each spectral line (which appear in any experiment I've run, for other pairs of numbers as well).
And how to find the size of the 'line broadening'? What does it depend on?
This is what the distribution looks like for terms which can take values $1,2,3$:
See also the thesis about nested radicals and Cantor sets here