Research Article

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

Published in: Quaestiones Mathematicae
Volume 48 , issue 8 , 2025 , pages: 1247–1260

DOI: 10.2989/16073606.2025.2478908

Author(s): Jiuzhou Zhao East China Normal University, China , Ruofan Li Jinan University, China

Keywords: Primary: Primary: 11A63 , Secondary: Secondary: 11R04 , Radix representation , digital problems , p-adic interpolation

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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 C ₁ N^σ ^(β), where and C ₁ is a constant depending only on β, if all prime ideal factors of β are unramified and their norms are integer primes.