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

TL;DR

This paper introduces a new binary-encoded tree structure called the V tree, which allows for exponentially faster coverage of rational numbers and extends Minkowski's question mark function into a binary setting, with conjectures on its differentiability.

Contribution

The paper defines the V tree using binary encodings adapted to the Gauss-Kuz'min measure, providing a more efficient way to enumerate rationals and extending Minkowski's question mark function into a binary framework.

Findings

Numbers with denominator q appear in the V tree within approximately 3.44 log2(q) levels.

The V tree covers all rationals exactly once.

The binary Minkowski question mark function ?_V is conjectured to have no derivative at rational points.

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 Gau\ss-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, $?'(x)=0, x\in Q$).