Re:Powers of 2 in lexical order yield decibels (2019)
by Someone on Tuesday March 01, 2022 @12:30PM
(#30517066)
I’m not sure it is with any starting point. 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)
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Re:Powers of 2 in lexical order yield decibels (2019)
by contravariant on Tuesday March 01, 2022 @02:32PM
(#30518891)
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.