This talk studies the map F(x) = x
ceiling(x).
For each rational r=p/q, with r >1, is it true
that some iterate of
r under F is an integer? The answer,
concerning the exceptions, is:
``No, never? Well, hardly ever.''
Similar questions are
considered for the ``approximate multiplication'' map
F_{r}(x) = r ceiling(x),
where r is a fixed rational number.
Starting from an initial value x_{o}, is it true that
some iterate of F_{r}(x) is an integer?
The talk will explain analogies with the notorious 3x+1
problem. This is joint work with Neil Sloane (AT&T Labs);
see math.NT/0309389.
