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Digit expansions in rational and algebraic basis

Lucía Rossi

TL;DR

The paper develops α-expansions for bases $\alpha\in\mathbb{Q}(i)$ with $|\alpha|>1$, using a lattice $\Lambda_\alpha$ and a backward-division algorithm to obtain integer expansions in the digit set $\mathcal{D}=\{0,\ldots,|a_0|-1\}$. It connects these algebraic number systems to tilings of the complex plane, establishes a finiteness property via shift radix systems, and extends the framework to all complex numbers with unique almost-everywhere expansions through a tiling argument; it further refines the theory with $p$-adic completions and ambinumber spaces $\mathbb{C}\times K_{den(\alpha)}$ to obtain a robust uniqueness result for expansions. The results cover both rational and algebraic bases, provide constructive algorithms for addition, multiplication, and approximation, and unify the geometric, combinatorial, and number-theoretic aspects through tilings and $p$-adic methods. The work opens avenues for generalizations to higher-degree bases, non-UFD settings, and questions about the structure and distribution of digits in complex expansions, with potential applications to complex-base numeration and fractal tilings. Overall, it advances a cohesive theory of complex-base expansions grounded in algebraic, geometric, and $p$-adic techniques, and clarifies when and how unique representations arise almost surely.

Abstract

Consider $α\in \Q(i)$ satisfying $|α| >1$. Let $\D = \{0,1,\ldots,|a_0|-1\}$, where $a_0$ is the independent coefficient of the minimal primitive polynomial of $α$. We introduce a way of expanding complex numbers in base $α$ with digits in $\D$ that we call $α$-expansions, which generalize rational base number systems introduced by Akiyama, Frougny and Sakarovitch, and are related to rational self-affine tiles introduced by Steiner and Thuswaldner. We define an algorithm to obtain the expansions for certain Gaussian integers and show results on the language. We then extend the expansions to all $x \in \C$ (or $x \in \R$ when $α= \ab \in \Q$, the rational case will be our starting point) and show that they are unique almost everywhere. We relate them to tilings of the complex plane. We characterize $α$-expansions in terms of $p$-adic completions of $\Q(i)$ with respect to Gaussian primes.

Digit expansions in rational and algebraic basis

TL;DR

The paper develops α-expansions for bases with , using a lattice and a backward-division algorithm to obtain integer expansions in the digit set . It connects these algebraic number systems to tilings of the complex plane, establishes a finiteness property via shift radix systems, and extends the framework to all complex numbers with unique almost-everywhere expansions through a tiling argument; it further refines the theory with -adic completions and ambinumber spaces to obtain a robust uniqueness result for expansions. The results cover both rational and algebraic bases, provide constructive algorithms for addition, multiplication, and approximation, and unify the geometric, combinatorial, and number-theoretic aspects through tilings and -adic methods. The work opens avenues for generalizations to higher-degree bases, non-UFD settings, and questions about the structure and distribution of digits in complex expansions, with potential applications to complex-base numeration and fractal tilings. Overall, it advances a cohesive theory of complex-base expansions grounded in algebraic, geometric, and -adic techniques, and clarifies when and how unique representations arise almost surely.

Abstract

Consider satisfying . Let , where is the independent coefficient of the minimal primitive polynomial of . We introduce a way of expanding complex numbers in base with digits in that we call -expansions, which generalize rational base number systems introduced by Akiyama, Frougny and Sakarovitch, and are related to rational self-affine tiles introduced by Steiner and Thuswaldner. We define an algorithm to obtain the expansions for certain Gaussian integers and show results on the language. We then extend the expansions to all (or when , the rational case will be our starting point) and show that they are unique almost everywhere. We relate them to tilings of the complex plane. We characterize -expansions in terms of -adic completions of with respect to Gaussian primes.

Paper Structure

This paper contains 14 sections, 17 theorems, 89 equations, 5 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction.
  2. Outline and main results
  3. Rational base number systems
  4. Algebraic number systems
  5. Finiteness property and shift radix systems
  6. The language of integer $\alpha$-expansions
  7. Algorithms for addition and multiplication
  8. Expansions of complex numbers
  9. Tilings for algebraic number systems
  10. $\alpha$-expansions and $p$-adic completions
  11. Ambinumbers
  12. Generalizations and open questions
  13. Generalizations
  14. Open questions

Key Result

Proposition 1

Let $\tfrac{a}{b} \in\mathbb{Q}$, $\tfrac{a}{b} <0$ with $-a > b > 2$, and $\mathcal{D} = \{0, \ldots, |a|-1\}$. Let $N = N_0 \in \mathbb{Z}$. Then the sequence $(N_j)_{j \geq 0}$ obtained through eq:algo is eventually zero.

Figures (5)

  • Figure 1: The tree $\mathcal{T}_\alpha$ for $\alpha = \frac{-1+3i}{2}$.
  • Figure 2: The tree $\mathcal{T}_\alpha$ for $\alpha = \frac{-1+5i}{3}$.
  • Figure 3: The tree $\mathcal{T}_\alpha$ for $\alpha = \frac{-1+5i}{3}$ but naming each node by the residue class of its children.
  • Figure 4: Tiles $\mathcal{G}_\alpha(N)$ for $\alpha = \frac{-1+3i}{2}$ and $N \in \Lambda_\alpha$
  • Figure 5: The set of series $\sum_{j \leq -1} \alpha^j d_j$ embedded in $\mathbb{C} \times K_{1+i}$.

Theorems & Definitions (51)

  • Proposition 1
  • proof
  • Definition 1
  • Theorem 1
  • proof
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 41 more