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