Probability distributions associated to the logarithm numeration system

Question

The most elementary logarithmic numeration system is defined as follow. Any random number $X \in [0, 1]$ can be represented uniquely as $$X=\log_3(A_1 + \log_3(A_2+\log_3 (A_3 + \cdots)))$$ with $A_k \in \{1, 2\}$ also a random variable if $X$ is a random variable. Let us introduce the sequence $X_n$ with $X_1=X$, as follows:

$$X_{n+1} = 3^{X_n}-A_n, \mbox{ with } A_n = 2 \mbox{ if } X_n\geq \log_3 2, \mbox{ and } A_n = 1 \mbox{ otherwise}.$$

The above formula allows you to compute the digits $A_1, A_2$ and so on. Also $X_n\in [0, 1]$. Assuming $X_1 = X$ is uniform on $[0, 1]$, I am interested in the following quantities:

• $p_n = P(X_n < \log_3 2)$, and especially $p_\infty$.
• $E(X_\infty^k)$, $k=1, 2, 3, 4$.
• The limiting distribution of $X_n$ and whether it admits a
  non-singular density function.
• $\lim_{n\rightarrow \infty}n\Big(E(X_n)-E(X_\infty)\Big)$

Any result, other than the ones I already discovered myself and listed in the next section, is welcome.

1. Theoretical results obtained so far

Probably the most fundamental theorem is this:

• If $X_n\geq \log_3 2$ then $P(X_{n+1}\geq \alpha) = \frac{1}{1-p_n}P(X_n\geq \log_3(2+\alpha))$
• If $X_n < \log_3 2$ then $P(X_{n+1}\geq \alpha) = \frac{1}{p_n}\Big[p_n -P(X_n\geq \log_3(1+\alpha))\Big]$

Thus $$P(X_{n+1}\geq\alpha) = p_n + P(X_n\geq \log_3(2+\alpha)) - P(X_n\leq \log_3(1+\alpha).$$ Here $\alpha\in[0, 1]$. Many simple results can be derived from this formula, in particular:

• $P(X_2\leq\alpha)=\log_3\Big[\frac{1}{2}(1+\alpha)(2+\alpha)\Big]$
• $p_2 = \log_3\Big[\frac{1}{2}(1+\log_3 2)(2+\log_3 2)\Big]$
• $E(X_2) = \frac{2+\log 2}{\log 3} - 2$

and more generally formulas like these ones, for $n>1$:

$$P(X_n <\alpha) = -p_{n-1} -(n-1)\log_3 2 + \log_3 \prod_{i_1,\cdots,i_{n-1}} B_{i_1,\cdots,i_{n-1}}$$

$$p_n = -(n-1)\log_3 2 + \log_3 \prod_{i_1,\cdots,i_{n-1}} C_{i_1,\cdots,i_{n-1}}$$

with (for instance)

• $A_{i_1,i_2,i_3} =\log_3(i_1+\log_3(i_2+(\log_3(i_3 +\alpha))), $
• $C_{i_1,i_2,i_3} =\log_3(i_1+\log_3(i_2+(\log_3(i_3 +\log_3 2)))$.

All indexes $i_1,i_2,i_3$ and so on take only on two values: $1, 2$.

In short all the quantities of interest can be computed recursively. Note that these results are from me, if you see any error or typo, please let me know.

2. Distribution of $X_{\infty}$

It is very well approximated by $P(X_\infty <\alpha) \approx \sqrt{\alpha}$. Note that $\alpha\in [0,1]$. Below are the empirical percentile distributions for $X_1$ (uniform), $X_2, X_3$ and $X_{40}$.

Below is the error between the empirical distribution of $X_{40}$ and its approximation based on a square root distribution on $[0, 1]$:

This a remarkable chart. I had expected to be fractal-like, highly chaotic as this is the case for the nested square root system, see here.

3. Empirical results

The chart below shows convergence of the first four moments $M_1,\cdots, M_4$, as well as that of $p_n$, starting with $X_1 = X$ being uniform on $[0, 1]$.

And here is some source code for the various computations:

$lg=log(2)/log(3);
$rand=sqrt(2)/2;
$m=40;           # X_1, ... X_m
$numbers=90000;  # sample size

open(OUT,">lognum.txt");

for ($k=0; $k<$numbers; $k++) {

  if ($k % 100==0) { print "$k\n"; }

  $x=$rand;
  $rand=3*$rand-int(3*$rand);  # uniform deviates
  $z=$x;

  for ($n=1; $n<=$m; $n++) {
    if ($z >= $lg ) { $digit=2; } else { $digit=1; } 
    $z2=$z*$z; ## to compute variance
    $z3=$z*$z2;
    $z4=$z*$z3;
    if ($n==40) { print OUT "$k\t$x\t$n\t$z\t$digit\t$z2\n"; }

    if ($digit==1) { $adigit[$n]++; }
    $az[$n]+=$z;
    $az2[$n]+=$z2;
    $az3[$n]+=$z3;
    $az4[$n]+=$z4;

    $z=3**$z - $digit;

  }

}
close(OUT);

open(OUT,">lognum2.txt");

Answer 1

Your theoretical result is very straightforward. If we have

$$X_{n+1} = 3^{X_n} - A_n$$

then

$$P[X_{n+1} \geq \alpha] = P[3^{X_n}-A_n \geq \alpha] = P[X_n \geq \log_3(\alpha + A_n)] $$

and you can split up the condition $X_n \geq \log_3(\alpha + A_n)$ into conditional cases when $X_n \geq \log_3(2)$ and when $X_n < \log_3(2)$. This is a split up in regions.

$$\begin{array}{} P[X_{n+1} \geq \alpha] &=& P[X_n \geq \log_3(\alpha + A_n)] \\ &=& P[2 > X_n \geq \log_3(\alpha + 1)] + P[ X_n \geq \log_3(\alpha + 2) \geq \log_3(2)] \\ &=& \underbrace{P[2 > X_n \geq \log_3(\alpha + 1)]}_{X_n<2} + \underbrace{P[ X_n \geq \log_3(\alpha + 2)] \hphantom{\geq \log_3(2)}}_{X_n \geq 2} \end{array}$$

---

It might be interresting to consider the following continued logarithm:

$Y_n = \log_3(B_1 + \log_3(B_2 + \log_3(B_3 + ....)))$

where $B_i = A_i \text{ if } i \leq n$ and $B_i = 1 \text{ if } i > n$

and $Y_\infty$ has all $B_i = A_i$ and equals $X$.

The $Y_n$ are like the $Y_\infty$ but with the digits above $n$ removed. Your $X_n$ is an expression for the difference between $Y_n$ and $Y_\infty$, but taken to the power 3 repeated $n$ times.

Then $$X_n = 3^{3^{3^{\dots (Y_{\infty})}}}-3^{3^{3^{\dots (Y_{n})}}} = \log_3(A_{n+1}+\log_3(A_{n+2}+\log_3(A_{n+3}+ \dots)))$$

---

Inspired by that you could also find the coordinates $A_i$ or $B_i$ by slicing the section $[0,1]$ into pieces according to numbers $Y_n$

It seems like the probability for the digit $P(A_n = 2)$ (the gray sections) decreases as $n$ is increasing, but the rate of decrease is slowing down for larger $n$.

For $n=16$, I computed the $Y_n$ for all possible $A_1, ... , A_n$. Based on that I computed the first $p_n$ and this gave on a log-log plot the following:

Do we have $\lim_{n \to \infty} p_n = 0$ ?

What we see is that everytime these sections are getting split up into a black and a gray section. What is the fraction of gray for each split?

I am taking some jump and pose that

• the right section is the one that gets split into the largest part gray. The point of split $z_n$ follows

  $$z_{n+1} = \log_3(2+z_n)$$

  with $z_1 = \log_3(2)$

  The fraction of gray is $f(z_n) = (1-z_{n+1})/(1-z_n)$. This gives more generally as a function of $q=1-z$

  $$f(q) = \frac{1-\log_3(3-q)}{q} \underbrace{ \approx \frac{\frac{1}{\log(27)}q }{q} }_{\text{Taylor series approximation}} \underbrace{\approx \frac{1}{\log(27) }\approx 0.3034}_{\lim q \to 0}$$

• the left section is the one that gets split into the smallest part gray. The point of split $z_n$ follows

  $$z_{n+1} = \log_3(1+z_n)$$

  with $z_1 = \log_3(2)$

  The fraction of gray is $f(z_n) = 1 - z_{n+1}/z_n$. This gives more generally as a function of $q=z$

  $$f(q) = 1 - \frac{\log_3(1+q)}{q} \underbrace{ \approx 1- \frac{\frac{1}{\log(3)}q }{q} }_{\text{Taylor series approximation}} \underbrace{\approx 1-\frac{1}{\log(3) }\approx 0.0898}_{\lim q \to 0}$$

So we get

$$ 0.0897 < p_{\infty} < 0.3035 $$

The same technique can be used to refine the bounds by going further down the tree instead of just using the left most and right most sections. For the moment I do not see a more intelligent way to find bounds for $p_\infty$