On digit patterns in expansions of rational numbers with prime denominator

TL;DR

This paper investigates the frequency of digit patterns in the base-$g$ expansion of fractions with prime denominators, showing that almost all such expansions contain nearly all possible strings of certain lengths, extending previous results.

Contribution

It extends known bounds on digit pattern occurrences in expansions of fractions with prime denominators, providing a broader understanding of their digit distribution.

Findings

Almost all fractions with prime denominator contain nearly all digit strings of certain lengths.

The length of digit strings covered is proportional to $rac{5}{24} imes ext{log}_g p$, improving previous bounds.

Results hold for almost all primes and fractions coprime to the denominator.

Abstract

We show that, for any fixed $\varepsilon > 0$ and almost all primes $p$, the $g$-ary expansion of any fraction $m/p$ with $\gcd(m,p) = 1$ contains almost all $g$-ary strings of length $k < (5/24 - \varepsilon) \log_g p$. This complements a result of J. Bourgain, S. V. Konyagin, and I. E. Shparlinski that asserts that, for almost all primes, all $g$-ary strings of length $k < (41/504 -\varepsilon) \log_g p$ occur in the $g$-ary expansion of $m/p$.