V Tree -- Continued Fraction Expansion, Stern-Brocot Tree, Minkowski's $?(x)$ Function In Binary: Exponentially Faster

Abstract

The Stern-Brocot tree and Minkowki's question mark function $?(x)$ (or Conway's box function) are related to the continued fraction expansion of numbers from Q with unary encoding of the partial denominators. We first define binary encodings $C_I,C_{II}$ of the natural numbers, adapted to the Gauß-Kuz'min measure for the distribution of partial denominators. We then define the V${}_1$ tree as analogue to the Stern-Brocot tree, using the binary encondings $C_I,C_{II}$. We shall see that all numbers with denominator $q$ are present in the first $3.44{\log }_2(q)$ levels, instead of $1/q$ appearing in level $q$ in the Stern-Brocot tree. The extension of the V${}_1$ tree, the V tree, covers all numbers from Q exactly once. We also define the binary version of Minkowski's question mark function, ${?}_V$, and conjecture that it has no derivative at rational points (for the original, ${?}^{\prime }(x)=0,x\in Q$).