See the final sections of the beautiful paper by L. R. Ford, recently mentioned elsewhere on math-fun --- https://www.maths.ed.ac.uk/~v1ranick/papers/ford.pdf WFL On 3/22/19, Dan Asimov <dasimov@earthlink.net> wrote:
Let z be any complex number whose real and imaginary parts are not both rational.
Set
z_0 = floor(Re(z)) + i*floor(Im(z))
(the lower left corner of the integer grid square that z lies in). Thereafter set
z_(n+1) = floor(Re(1/z_n)) + i*floor(Im(1/z_n))
Then z_0, z_1, ..., z_n, ... in Z[i] form a series of Gaussian integers representing z, generalizing continued fraction expansions of irrational real numbers.
* Have such things been studied?
* For almost all real numbers x, the geometric mean of all the "digits"*
gm(x) = lim (K_0 K_1 ... K_n)^(1/n) n —> oo
of the cfe of x is equal to the same number. What about the corresponding geometric mean of the Gaussian integers
gm(z) = (z_0 z_1 ... z_n)^(1/n) ???
—Dan ————— * Should we call them "tigids"?
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