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Four restrictions


Notes:

In order to get a relatively simple and clear formulation of the theorem, I had to make the four restrictions (SHOW THEM). Why?:

a<0 – you have to go backwards in the numeration of the objects (symmetry about y-axis solves the problem)
a>1 – you do not deal with a function Z->Z (symmetry about the diagonal y=x solves the problem)
a rational – you get a lot of special cases; there is only a finite number of digitization levels, you get periodicity.
The problem for rational slopes has a nice complete solution with application of the euclidic algorithm (I showed this a year ago during my presentation here).
x<0 – you have to make the calculations with negative integers (a kind of point symmetry solves the problem).