On digit patterns in expansions of rational numbers with prime denominator

2012/05/25 by Igor E. Shparlinski, Shparlinski, Igor E., Wolfgang Steiner +1
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1205.5673

arxiv created 2012/05/25 · openalex publication_date 2012/05/25 · arxiv updated 2012/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that, for any fixed ε > 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 - ε) 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 -ε) log_g p occur in the g-ary expansion of m/p.

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