The repetends of reduced fractions $a/b^k$ approach full complexity with an increasing $k$

TL;DR

This paper establishes a criterion for the complexity of $g$-ary expansions of rational fractions and shows that as $k$ increases, the digit sequences in the repetends of $a/b^k$ become uniformly distributed, approaching full randomness.

Contribution

It provides a new criterion for complexity in $g$-ary expansions of fractions and demonstrates the asymptotic uniform distribution of digit sequences in the repetends of $a/b^k$ as $k$ grows.

Findings

Digit sequences in $a/b^k$ approach uniform distribution as $k$ increases.

Absolute frequencies of digit sequences can be computed using a transition matrix.

Uniform distribution fails when all prime factors of $b$ divide the base $g$.

Abstract

In this paper, we prove a criterion for complexity in $g$-ary expansions of a rational fraction $a/b<1$ with gcd$(a,b)=1$. We prove that for any purely periodic proper fraction $a/b$ and all $j\geq 1$, each sequence of $j$ digits occurs in the $g$-ary repetend of $a/b^k$ with a relative frequency that approaches $1/g^j$ with an increasing $k$. The absolute frequencies can be calculated by means of a simple transition matrix. Let $(a_k)$ be a sequence of positive integers relatively prime to $b$. We prove that each sequence of $j$ digits occurs in the $g$-ary repetend of $a_k/b^k$ with a relative frequency that approaches $1/g^j$ with an increasing $k$, unless all prime factors of $b$ divide the base $g\geq 2$.