On digit patterns in expansions of rational numbers with prime denominator

Abstract

We show that, for any fixed $\epsilon >0$ 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 $k<(5/24-\epsilon ){\log }_{g}p$. 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 $k<(41/504-\epsilon ){\log }_{g}p$ occur in the $g$-ary expansion of $m/p$.