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arXiv · 1205.5673

On digit patterns in expansions of rational numbers with prime denominator

Igor E. Shparlinski·Wolfgang Steiner
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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 $m/p$.

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Igor E. Shparlinski, Wolfgang Steiner. 2012-05-25. On digit patterns in expansions of rational numbers with prime denominator. https://arxiv.org/abs/1205.5673

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