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Cobham's theorem for the Gaussian integers

Álvaro Bustos-Gajardo, Robbert Fokkink, Reem Yassawi

TL;DR

This paper generalizes Cobham's theorem to Gaussian integers, showing that for multiplicatively independent Gaussian bases α and β with |α|,|β|>1, an α- and β-automatic configuration is eventually periodic whenever at least one base is not a root of an integer. It develops Gaussian numeration systems, a Dirichlet-approximation-type tool in Z[i], and a period-transfer mechanism that propagates local periodicity to the entire Gaussian lattice via geometric ball coverings, avoiding reliance on the four exponentials conjecture. The results yield a Semenov-type, higher-dimensional Cobham–Semenov theorem for Euclidean domains and confirm Hansel and Safer's conjecture in the non-root case, while showing sharpness through counterexamples when both bases are roots of integers.

Abstract

Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $α=-m+i $- and $β=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually periodic. Without assuming the four exponentials conjecture, we show that if $α$ and $β$ are multiplicatively independent Gaussian integers, and at least one of $α$, $β$ is not an $n$-th root of an integer, then any $α$- and $β$-automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are $α$-automatic for any root of an integer $α$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations.

Cobham's theorem for the Gaussian integers

TL;DR

This paper generalizes Cobham's theorem to Gaussian integers, showing that for multiplicatively independent Gaussian bases α and β with |α|,|β|>1, an α- and β-automatic configuration is eventually periodic whenever at least one base is not a root of an integer. It develops Gaussian numeration systems, a Dirichlet-approximation-type tool in Z[i], and a period-transfer mechanism that propagates local periodicity to the entire Gaussian lattice via geometric ball coverings, avoiding reliance on the four exponentials conjecture. The results yield a Semenov-type, higher-dimensional Cobham–Semenov theorem for Euclidean domains and confirm Hansel and Safer's conjecture in the non-root case, while showing sharpness through counterexamples when both bases are roots of integers.

Abstract

Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset of the Gaussian integers is both - and -recognizable, then it is syndetic, and they conjectured that must be eventually periodic. Without assuming the four exponentials conjecture, we show that if and are multiplicatively independent Gaussian integers, and at least one of , is not an -th root of an integer, then any - and -automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are -automatic for any root of an integer . Our work generalises the Cobham-Semenov theorem to Gaussian numerations.

Paper Structure

This paper contains 8 sections, 19 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gaussian numerations
  4. Dirichlet approximation in Z[i]
  5. Finite automata and automaticity
  6. Generating periods
  7. Periodic patterns on Z[i]
  8. Proof of the main theorem for Z[i]

Key Result

Theorem 1

Let $\alpha$ and $\beta$ be two multiplicatively independent Gaussian integers with $|\alpha|, |\beta|>1$. If one of $\alpha$ or $\beta$ is not the root of an integer, then a configuration $(a_z)_{z\in \mathbb Z[i]}$ is $\alpha$- and $\beta$-automatic if and only if it is eventually periodic.

Figures (2)

  • Figure 1: A $(1+2\mathrm{i})$-automatic configuration with digit set $D=\{0,\pm 1,\pm \mathrm{i}\}$.
  • Figure 2: A configuration with step-period $(1+4\mathrm{i},1-4\mathrm{i})$ on the ball of radius $5 + \epsilon$, for some $\epsilon <1$, around the origin that does not satisfy the definition of period. The values $z=-5$ and $z=1$ are congruent modulo the lattice $\langle 1+4\mathrm{i},1-4\mathrm{i}\rangle$, so the definition of periodicity requires them to be the same color; however, all four neighbours $5\pm(1\pm 4\mathrm{i})$ fall outside the ball, so step-periodicity is not enough to guarantee that they do actually have the same color.

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Lemma 3: Bosma-Fokkink-Krebs, Lemma 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • Theorem 6
  • proof
  • Definition 7
  • ...and 28 more