Bibliography request: Entropy for continued fractions

Given a strictly positive real number $$x$$ we set $$e(x)=\log(1+x)$$ if $$x$$ is an integer and $$e(x)=\log(1+x)+\frac{1+\lbrace x\rbrace}{1+x}\left(e(1/\lbrace x\rbrace)-\log(1+\lbrace x\rbrace)\right)$$ otherwise (i.e. if the fractional part $$\lbrace x\rbrace$$ of $$x$$ is strictly positive).

This defines $$e(x)$$ if $$x$$ is rational and the recursive definition of $$e(x)$$ yields its value as a (perhaps infinite) limit if $$x$$ is irrational.

The real number $$e(x)$$ can be interpreted as an entropy (or information content) as follows: Consider the segment $$[-1,x]$$ endowed with the uniform probability law. Apply to the positive and to the negative part of this segment iteratedly the Euclidean Algorithm (removing from the longer piece a segment of length the length of the shorter piece). This yields a partition of $$[-1,x]$$ into finitely many segments if $$x$$ is rational and into infinitely many segments accumulating at the origin otherwise. The function $$e(x)$$ is simply a normalized entropy of this partition ($$e(x)$$ is essentially the expected amount of information on the position of a random point on $$[-1,x]$$ which can be inferred when only knowing the interval containing the random point).

We illustrate this with $$x=5/19$$. We start with the interval $$[-1,5/19]$$. Since only proportions matter, we replace it by $$[-19,5]$$ in order to work over $$\mathbb Z$$. The positive interval $$[0,5]$$ is shorter than the negative interval $$[-19,0]$$. We therefore cut a piece of length $$5$$ from the negative interval in order to get $$[-14,5]$$. Iteration yields then $$[-9,5]$$ and $$[-4,5]$$. At this point the negative interval is shorter and we get therefore $$[-4,1]$$. The positive interval $$[0,1]$$ is now shortest leading to $$[-3,1],[-2,1],[-1,1]$$ and finally $$[ 0,1]$$. We get therefore the partition of $$[-19,5]$$ into intervals $$[-19,-14],[-14,-9],[-9,-4],[-4,-3],[-3,-2],[-2,-1],[-1,0],[0,1],[1,5]$$. Endowing the interval $$[-19,5]$$ with the uniform probability measure $$\mu$$ (with probability of an interval given by its length divided by the total length $$19+5=24$$) the topological entropy of this partition is given by $$e(5/19)=-\frac{1}{24}(5\log(5/24)+5\log(5/24)+5\log(5/24)+5\cdot 1\log(1/24)+4\log(4/24))$$ and gives the expected amount of information (up to $$\log(2)$$ when working with bits, I guess) encoded by the partition.

A few easy facts:

$$e(x)=e(1/x)$$ (obvious from the definition).

$$e(x)$$ is discontinous at every rational value (it is in fact unbounded in the neighbourhood of a rational value) but should be continuous on a set of full measure. $$e(x)$$ is infinite at some irrational values with very huge coefficients in their continued fraction expansion.

The graph of $$e(x)$$ is a beautiful fractal.

The function $$x\longmapsto e(x)$$ is probably locally integrable.

It would be somewhat surprising if nobody has ever looked at this notion.

Are there paper, references etc. on this notion?

• Could you be more specific concerning the definition of the partitions you are talking about? What is this partition already for an integer $x$? What do you mean by "a normalized entropy"?
– R W
Oct 22, 2022 at 14:45
• @RW Thanks for your input. I have added a hopefully helpful example. Oct 22, 2022 at 15:12
• Thank you - it's clearer now. Since you have already mentioned continued fractions, could you describe your partition directly in terms of the contiued fraction decomposition of $x$?
– R W
Oct 22, 2022 at 15:51
• @RW yes, I can but it is slightly indirect and involves all convergents and semi-convergents. This question arose from this description but I wanted it to be not too long. Oct 22, 2022 at 16:05

Below is a graph of $$e(x)$$. Precisely, the $$y$$-value associated with the given $$x$$-value is $$y = e(x/70009)$$.