3 ms·

I’m not sure it is with any starting point. https://mathworld.wolfram.com/PowerFractionalParts.html https://mathworld.wolfram.com/PowerFractionalParts.html desc

I’m not sure it is with any starting point. https://mathworld.wolfram.com/PowerFractionalParts.html https://mathworld.wolfram.com/PowerFractionalParts.html describes a similar sequence, of which Hardy and Littlewood proved it is equidistributed for almost all real numbers (and the exceptions aren’t all trivial. The golden ratio and 1+√2 are exceptions, for example)

contravariant 5y ago
It is, but you're right to question it because it's a very nontrivial property that makes irrational rotations a bit of a special case. The fact that it's uniquely ergodic (which roughly means that the uniform Lebesgue measure is the only measure that's invariant under irrational rotation) is required to ensure it works for all starting points. Most ergodic systems only ensure that it works for almost all starting points i.e. all starting points except for a measure 0 set.