On β-adic expansions of powers of algebraic integer omitting a digit

2024/05/10 by Jiuzhou Zhao, Ruofan Li, Zhao, Jiuzhou +1
Mathematics · #advanced mathematical theories #Mathematical Dynamics and Fractals #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2405.06220

Abstract

Let α, β be two relatively prime algebraic integers in a number field K and N be a positive integer. We show that the number of n∈\1,2,…,N\ such that the β-adic expansion of αⁿ omits a given digit is less than C₁ N^σ(β), where σ(β):=(log(|N(β)|-1))/(log|N(β)|) and C₁ is an absolute constant, if all prime ideal factors of β are unramified and their norms are integer primes.

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