What is the distribution of real numbers with biased digits? Suppose I have an infinite sequence of biased bits where the probability of $1$ is $2/3$ and the probability of $0$ is $1/3.$ If I view these as the digits in the binary expansion of a real number, then this sequence defines a real number in the interval $[0,1]$. So what kind of distribution does this real number have?
Some considerations I have made so far is that the probability between $0.5$ and $1$ should be twice the probability between $0$ and $0.5.$ Similarly the probability between $0.25$ and $0.5$ should be twice the probability between $0$ and $0.25.$ A general way of writing this is recursive relationship is
$$F(2x) - F(x) = 2F(x).$$
Adding boundary conditions I get the three equations
$$F(0)=0\\
F(1)=1\\
F(2x)=3F(x)$$
which, if viewed as a recurrence relation, has the solution $F(x) = x^{\log_2(3)}$. My question is: Is this really airtight? Setting up these equations and using the solution from a recurrence relation felt a little hand wavy. I can easily verify that $x^{\log_2(3)}$ satisfies the above conditions for real numbers in the interval $[0,1]$, but is this solution unique?

Your recursion gives a condition the cumulative distribution function satisfies (in a sense, you have fractal copies of the function in itself), but there are several functions which satisfy this.
You would not expect the cumulative distribution function to be a smooth function since for example values of the binary form $0.0111xyz\ldots_2$ are four times as likely as those of the form $0.1000xyz\ldots_2$
The cumulative distribution function seems to look like this red line while $x^{\log_2(3)}$ is the blue line

and you can see that $x^{\log_2(3)}$ only gives the correct value when $x$ is a negative power of $2$, as Ross Millikan commented
When $x=\dfrac{k}{2^n}$ for some integers $k,n$, you have $F(x)=\dfrac{a(k)}{3^n}$ where $a(k)$ is OEIS A006046 (the number of odd entries in the first $k$ rows of Pascal's triangle). Other values can be found by limits since $F(x)$ is increasing, and looking at Michael Hardy's example it seems that you should have  $F(\frac15)=\frac{5}{77},\, F(\frac25)=\frac{15}{77},\, F(\frac35)=\frac{29}{77},\, F(\frac45)=\frac{45}{77}$  
