Consider the bit-representation of some irrational number, for instance
$$\alpha=\sqrt2=\text{"1.011010100000100111100110011…"}_2 .$$
If I do the partial approximations with left $n$ binary digits then I'm always below the true value. It means the partial approximations $$a_n= \sum_{k=0}^n \text{dig}(k)2^{-k} \lt \alpha$$ But it is easy to see that introducing digits from $\{-1,0,1\}$ allows partial approximations which are nearer than the approximations before.
I fiddled a bit with the programming of such representation, but I'm unsure, whether this representations are unique.
Example let "i" denote the $-1$-digit. If I replace the leading bitstring $\text{"1.011"}$ by $\text{"1.10i"}$ I replace the sequence of approximations $1,1,5/4,11/8$ by $1,3/2,3/2,11/8$ which gives the sequences of errors as
$$ \small \begin{array}{rrrr}[&0.414214,& 0.414214,& 0.164214,& 0.0392136,&…] \\\
[&0.414214,& -0.085786,& -0.085786,& 0.0392136,&…]
\end{array}
$$
I can now in general replace consecutive equal digits of the form $\text{"x011..110…"}$ by $\text{"x100..0i0…"}$. But it happens, that the introduction of the first $\text{"1"}$ concatenates with the previous $\text{"1"}$ and I would have to repeat the process until no more change occurs.
This throws now the problem of uniqueness:
Can I assume that this (repeated) process gives the best partial approximations to the irrational target?
Remark: I'm insure about the best tags for the question. Please feel free to improve
Best Answer
It seems this goes towards a unique solution, which is distinguished (or "unique") by the property that no two nonzero digits can follow each other.
For the $\sqrt2$ I get by this a balanced binary representation as
"1.10i010100000101000i010i010i01000000i01000i000i010i001001000..."It looks like I could use the "balanced quaternary" representation instead whose digits are obvious when grouping the above string in 2-digits:
"1.10 i0 10 10 00 00 10 10 00 i0 10 i0 10 i0 10 00 00 0i 01 00 0i 00 0i 01 0i 00 10 01 00 ..."giving (using
jfor negative2)"1.2 j 2 2 0 0 2 2 0 j 2 j 2 j 2 0 0 i 1 0 i 0 i 1 i 0 2 1 0 ..."_{balanced4}The "moving best-partial-approximations" occur when a truncation follow many zeros:
"1.10i010100000101000i010i010i01000000i01000i000i010i001001000..."------------------------------------------------------------------"1.1""1.10i0101""1.10i010100000101""1.10i010100000101000i010i010i01"which means, $w(c)=\sqrt 2 \cdot 2^c$ for $c\in\{1,7,29,334,... \}$ approximate especially well to integers, each one better than all $w(c)$ before.
So this "indexing" by the binary representation of the number agrees with the simple numerical checks computing the fractional parts of $w(c)$ directly and seems to be a meaningful version of a "balanced-binary" representation with the desired property of exposing the best partial approximations as asked for in the title.