Actually, it isn't so easy, even for the Gaussian integers! What you need is to define a tiling of the Gaussian plane with tiles that fit within circles. But these tiles don't have to be square, and they don't have to have straight-line boundaries. E.g., you could have tiles with fractal boundaries. Many theorems work even if the tiles aren't exactly the same, so long as there is a relatively easy way to compute to which tile a given point belongs, and so long as the tiles fit within the circles. Check out http://home.pipeline.com/~hbaker1/Gaussian.html for some discussion of these issues. For some of the other Euclidean domains, such as the one you mention, you can scale one of the axes so that you can still use circles. See Hardy & Wright. At 01:21 PM 10/8/03 +1300, Mike Stay wrote:
Are there any useful definitions of greatest integer function for algebraic integers? For the Gaussian integers, it's pretty straightforward because it uses the Euclidean norm--find the nearest integer to a complex x with norm less than magnitude(x). It's unique except for some points on diagonals where there are two.
What about Z[sqrt(-3)]? I guess one could talk about integers with norm less than magnitude(x), but the magnitude of x is using a Euclidean norm while Z[sqrt(3)] uses a hyperbolic norm,
N(a+b sqrt(-3)) = a^2-3b^2.
(I think I got that right.) When the greatest integer isn't unique, can one show there are only a finite number of possibilities?
-- Mike Stay staym@clear.net.nz http://www.xaim.com/staym