Dan writes:
>A folk theorem holds that the vector defined by
>
> V_n := (phi,phi^2,...,phi^n),
>
>where p is the golden ratio, can't be beaten, at least in the limit.
I don't understand this. Even if we just look at the first three
components of the vector, we see that [m phi^3] (which I'll remind
everyone means the fractional part of m phi^3, under Dan's notation)
is equal to [m phi^2] plus [m phi], for all m, so the set of multiples
of (phi,phi^2,phi^3) mod (1,1,1) stays in the part of the torus (x,y,z)
with x+y=z mod 1, and hence cannot be dense in T^3. Or am I missing
something?
Leaving that quibble aside, I like Dan's geometric point of view.
Even if (phi,phi^2,...,phi^n) is the wrong vector to use, some
vectors are going to be okay. And Dan's concatenation method is
bound to give something decent even if it doesn't give N/log N.
Thanks, Dan!
Jim
