On $\beta$-adic expansions of powers of algebraic integer omitting a digit

TL;DR

This paper investigates the frequency of omitted digits in the $eta$-adic expansions of powers of algebraic integers, establishing an upper bound related to the algebraic properties of $eta$ and the number of such powers.

Contribution

It provides a new upper bound on the count of powers with $eta$-adic expansions missing a digit, under specific algebraic and ramification conditions.

Findings

Number of powers with omitted digit grows slower than $N^{\sigma(eta)}$

Bound depends on the norm of $eta$ and ramification conditions

Establishes a link between algebraic properties and digit omission frequency

Abstract

Let $\alpha, \beta$ be two relatively prime algebraic integers in a number field $K$ and $N$ be a positive integer. We show that the number of $n\in\{1,2,\dots,N\}$ such that the $\beta$-adic expansion of $\alpha^n$ omits a given digit is less than $C_1 N^{\sigma(\beta)}$, where $\sigma(\beta):=\frac{\log(|N(\beta)|-1)}{\log|N(\beta)|}$ and $C_1$ is an absolute constant, if all prime ideal factors of $\beta$ are unramified and their norms are integer primes.