Finite precision lexicographic continued fraction number systems

Lexicographic continued fraction binary (LCF) representation provides an order preserving bitstring representation of the non negative real numbers where every rational number has a finite length bitstring representation. We investigate the precision of k-bit LCF approximation. The maximum gap size over [0,1] for (k+1)-bit LCF representation is shown to be less than 2^−.81k, comparable to binary coded decimal in worst case representation efficiency. The distribution of gap sizes for (k+1)-bit LCF representation over [0,1] is shown on a logarithmic scale to be bell shaped between 2^−.81k and 2^−1.39k, becoming more peaked near the value corresponding to uniform spacing, 2^−k, with increasing k.