There are many papers on this subject. I wrote about it in my undergraduate thesis (available at https://cs.uwaterloo.ca/~shallit/papers.html, under "1979"). My thesis contains references to other papers. It turns out that your particular suggestion of how to do complex floor is not a good one, but other choices work. On 3/21/19 9:01 PM, Dan Asimov 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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