3 ms·
For all intents and purposes multiplying by 2 and multiplying it by a power of 10 until it's between 1 and 10 is equivalent to the dynamical system called an ir
by contravariant 5y ago
For all intents and purposes multiplying by 2 and multiplying it by a power of 10 until it's between 1 and 10 is equivalent to the dynamical system called an irrational rotation.
This equivalence isn't particularly deep but what is important is that irrational rotations are uniquely ergodic. This means that for all intents and purposes this procedure with any starting point will give a set of points that uniformly cover the unit interval (or in this case 1 to 10, logarithmically).
It also helps that 2^10 is a near miss for 10^3.
- Someone 5y agoI’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 agoIt 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.