It is well known that the sequence $\{2^k \eta\} \bmod 1$ is uniformly distributed for almost all, but not all, irrational $\eta$ in $(0,1)$. If I fix an irrational number $0<x<1$ ($x$ is actually of the following form $x=0.0100100101000000100001\dots$, i.e. many zeros and some ones), is it possible to say something about the distribution of $\{2^k x\}$ in $(0,1)$?
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Here is an interesting example I found in archives ... – rtybase May 19 '20 at 18:30
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Thank you @rtybase! Your (old) answer is very interesting, but I don't know about using the density argument for my issue. The key, I think, is that my number is random (in some sense), so I'm expecting a uniform distribution! – Canjioh May 19 '20 at 19:40
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As you say, it is uniformly distributed for almost every $x$ but not for every $x$. What you can say about a particular $x$ will depend on what you know about $x$. In particular, what is important is the base-$2$ expansion of $x$.
Robert Israel
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${2^k x}$ is obtained by shifting left the base-$2$ expansion $k$ places and removing the integer part. – Robert Israel May 19 '20 at 20:47
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Ok, I got it. In my problem I have many more zeros and their position is "random", I don't know if it helps, but thank you for your time! – Canjioh May 19 '20 at 23:00