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This is a problem which emerged from trying to prove that a sequence of random variables on $[0,1]$ does not converge almost surely to $0$.

In the process of trying to prove the above, I encountered the infinite sequence constructed as follows.

  • $x_1 = 1$
  • $x_2 = 1 - \frac{1}{2} - \lfloor (1 - \frac{1}{2}) \rfloor = \frac{1}{2}$, $\qquad \qquad$ i.e. the fractional part of $(1 - \frac{1}{2})$
  • $x_3 = 1 - \frac{1}{2} - \frac{2}{3} - \lfloor (1 - \frac{1}{2} - \frac{2}{3}) \rfloor = \frac{1}{2} + \frac{1}{3} $, $\qquad$ i.e. the fractional part of $(1 - \frac{1}{2} - \frac{2}{3})$
  • $\vdots$
  • $x_n = 1 - \sum_{i=2}^n \frac{n-1}{n} - \lfloor (1 - \sum_{i=2}^n \frac{n-1}{n}) \rfloor$, $\qquad$ i.e. the fractional part of $(1 - \sum_{i=2}^n \frac{n-1}{n})$
  • $\vdots$

I would like to know whether this sequence is dense in $[0,1]$. I have a strong intuition that it is, but I have no idea how to tackle the proof. Any idea on how I could prove or disprove it?

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    $\begingroup$ Fractional, rather than decimal. $\endgroup$
    – Andrés E. Caicedo
    Sep 2, 2014 at 22:51

1 Answer 1

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Hint: Denoting $\mathrm{frac}(x)$ to mean the fractional part of x, stating

$$ x_{i+1} = \mathrm{frac}\left( x_i - \frac{n-1}{n} \right) $$

is equivalent to stating

$$ x_{i+1} = \mathrm{frac}\left( x_i + \frac{1}{n} \right) $$

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    $\begingroup$ ... and so you are looking at whether the fractional parts of the Harmonic numbers are dense in $[0,1]$. The Harmonic numbers increase without limit, but with arbitrarily small increments, so ... $\endgroup$
    – Henry
    Sep 3, 2014 at 6:54
  • $\begingroup$ Many thanks, very good hint, that should be enough for me to figure out the rest of the argument. $\endgroup$
    – Martin Van der Linden
    Sep 3, 2014 at 12:23

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