2020/08/18 by Michael Vielhaber, Vielhaber, Michael
Computer Science · Physics and Astronomy · #Algorithms and Data Compression #FOS: Mathematics #Number Theory (math.NT) #Numerical Methods and Algorithms #Statistical Mechanics and Entropy #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2008.08020
openalex publication_date 2020/08/18 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
The Stern-Brocot tree and Minkowki's question mark function ?(x) (or\nConway's box function) are related to the continued fraction expansion of\nnumbers from Q with unary encoding of the partial denominators. We first define\nbinary encodings CI, CII of the natural numbers, adapted to the\nGau ss-Kuz'min measure for the distribution of partial denominators. We then\ndefine the V1 tree as analogue to the Stern-Brocot tree, using the binary\nencondings CI, CII. We shall see that all numbers with denominator q\nare present in the first 3.44\log2(q) levels, instead of 1/q appearing in\nlevel q in the Stern-Brocot tree. The extension of the V1 tree, the V\ntree, covers all numbers from Q exactly once. We also define the binary version\nof Minkowski's question mark function, ?V, and conjecture that it has no\nderivative at rational points (for the original, ?'(x)=0, x\∈ Q).\n