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

Jiuzhou Zhao, Ruofan Li

TL;DR

This paper extends the study of digits omission in radix-like expansions to β-adic expansions in algebraic number fields. It introduces the β-adic expansion as a p-adic-like representation relative to a fixed β and a residue system, and defines M_b(α,β,N) counting n where α^n omits a digit. Under the assumption that β factors into unramified prime ideals whose norms are rational primes, and gcd(α,β)=1, the authors prove M_b(α,β,N) ≤ C_1 N^{σ(β)} with σ(β) = log(|N(β)|-1)/log|N(β)|, generalizing Narkiewicz's bound. The method relies on p-adic interpolation of the sequence α^n, constructing analytic interpolants on p-adic unit balls and combining with a counting argument to derive the power-saving bound. The results extend to CNS contexts and provide a framework to study digit omission across number fields, potentially informing related persistence-like problems.

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 $n\in\{1,2,\dots,N\}$ such that the $β$-adic expansion of $α^n$ omits a given digit is less than $C_1 N^{σ(β)}$, where $σ(β):=\frac{\log(|N(β)|-1)}{\log|N(β)|}$ and $C_1$ is an absolute constant, if all prime ideal factors of $β$ are unramified and their norms are integer primes.

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

TL;DR

This paper extends the study of digits omission in radix-like expansions to β-adic expansions in algebraic number fields. It introduces the β-adic expansion as a p-adic-like representation relative to a fixed β and a residue system, and defines M_b(α,β,N) counting n where α^n omits a digit. Under the assumption that β factors into unramified prime ideals whose norms are rational primes, and gcd(α,β)=1, the authors prove M_b(α,β,N) ≤ C_1 N^{σ(β)} with σ(β) = log(|N(β)|-1)/log|N(β)|, generalizing Narkiewicz's bound. The method relies on p-adic interpolation of the sequence α^n, constructing analytic interpolants on p-adic unit balls and combining with a counting argument to derive the power-saving bound. The results extend to CNS contexts and provide a framework to study digit omission across number fields, potentially informing related persistence-like problems.

Abstract

Let be two relatively prime algebraic integers in a number field and be a positive integer. We show that the number of such that the -adic expansion of omits a given digit is less than , where and is an absolute constant, if all prime ideal factors of are unramified and their norms are integer primes.
Paper Structure (4 sections, 12 theorems, 65 equations)

This paper contains 4 sections, 12 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. $\beta$-adic expansion
  3. $\mathfrak{p}$-adic interpolation of the sequence $(\alpha^n)_{n\in\mathbb N}$
  4. Proof of Theorem \ref{['2402171658']}

Key Result

Theorem 1.5

Suppose $(\beta,\,\{0,1,\dots,|N(\beta)|-1\})$ is a CNS, $\beta$ is not divided by ramified primes and $\alpha$ is relatively prime to $\beta$, then holds for any digit $b\in\{1,\dots,|N(\beta)|-1\}$, where $\sigma(\beta):=\frac{\log(|N(\beta)|-1)}{\log|N(\beta)|}$ and $C_1$ is a constant depending only on $\beta$.

Theorems & Definitions (30)

  • Conjecture 1.1
  • Conjecture 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6: Kovács Kovacs1981
  • Definition 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Definition 2.1
  • Remark 2.2
  • ...and 20 more